1.Introduction

A. Bremner, R.K. Guy and R.J. Nowakowski[1] studied the diophantine equation (x+y+z)(1/x+1/y+1/z) = n
and gave the condition for the positive integer solution.
It is known as Melvyn Knight's problem.
Yong Zhang[2] studied the diophantine equation (x+y+z+w)(1/x+1/y+1/z+1/w) = n and gave the condition for the positive integer solution.
Andrew Bremner and Tho Nguyen Xuan[3] studied the diophantine equation (x+y+z+w)(1/x+1/y+1/z+1/w) = n and showed that there cannot exist
such positive solutions when n is of the form 4m^2, 4m^2+4, where m<>2 (mod 4).

We treat diophantine equation (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n. 
(x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = 25+(x-y)^2/(xy)+(x-z)^2/(xz)+(x-w)^2/(xw)+(x-s)^2/(xs)+(y-z)^2/(yz)+(y-w)^2/(yw)+(y-s)^2/(ys)+(z-w)^2/(zw)+(z-s)^2/(zs)+(w-s)^2/(ws)
Hence there is no positive integer solution for n<25.
We serached the positive integer solutions with 25<=n<10000.
As a result, diophantine equation (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n has a positive integer solution
for every integer n.
It seems that (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n has a positive integer solution for any n >=25.
We show the solution table for n<1000.
Furthermore, we show the solution of (x+y+y+w+w)(1/x+1/y+1/y+1/w+1/w) = n.
It seems that there are infinitely many positive integers n for (x+y+y+w+w)(1/x+1/y+1/y+1/w+1/w) = n.

(2018.7.26)
We extended the range up to 50000.
Diophantine equation (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n has a positive integer solution
for every integer n < 50000.
Solution table was added. Solutions:( 1000 <=n < 50000, zip file)


2.Thorem

Diophantine equation (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n has a parametric solution.

x = (b-1)(a-1)(a+b)
y = ab(a+b-1)(-1+n)
z = ab(-1+n)
w = -b(a+b-1)(-1+n)
s = -a(a+b-1)(-1+n)
a,b,n are arbitrary.

Proof.

(x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n........................................(1)

Multiplying xyzws to both side, we obtain
(zws+yws+yzs+yzw)x^2+(yw^2s+yzw^2+y^2ws+y^2zs+y^2zw+zw^2s+z^2ws+yz^2s+yz^2w+5*yzws+zws^2+yws^2+yzs^2-yzws*n)x
+yzw^2s+y^2zws+yzws^2+yz^2ws=0..............................................(2)

Let (y,z,w,s)=( ab(a+b-1)t, abt, -b(a+b-1)t, -(a+b-1)at ) then zws+yws+yzs+yzw becomes to 0.

Hence we obtain x = t(b-1)(a-1)(a+b)/(-1+n).

Let t=n-1, then x=(b-1)(a-1)(a+b).

Finally, we obtain a parametric solution.

Unfortunately, this solution doesn't give a positive integer solution.




3.Results

Smallest soutions are shown.
25<= n <1000

[  n    x   y   z   w   s]

[  25    1   1   1   1   1]
[  26    3   3   2   2   2]
[  27    2   1   1   1   1]
[  28    2   2   1   1   1]
[  29    6   5   4   3   2]
[  30    3   2   2   1   1]
[  31    4   3   2   2   1]
[  32   14   7   6   5   3]
[  33    3   3   1   1   1]
[  34    4   1   1   1   1]
[  35    4   3   3   1   1]
[  36    4   4   2   1   1]
[  37    5   2   1   1   1]
[  38    6   2   2   1   1]
[  39    6   3   2   1   1]
[  40   15  10   9   9   2]
[  41    5   5   3   1   1]
[  42    6   3   1   1   1]
[  43    6   5   2   1   1]
[  44    8   8   4   3   1]
[  45    8   8   6   4   1]
[  46    8   4   2   1   1]
[  47   21  15  10   8   2]
[  48   15  10   3   2   2]
[  49   12   4   2   2   1]
[  50    6   6   1   1   1]
[  51    9   9   6   2   1]
[  52    7   6   6   1   1]
[  53   12   7   6   2   1]
[  54   10   2   1   1   1]
[  55    8   8   2   1   1]
[  56   12   4   3   1   1]
[  57   11   6   3   1   1]
[  58    8   8   6   1   1]
[  59    9   6   1   1   1]
[  60   12   6   4   1   1]
[  61   14   3   2   1   1]
[  62   15  10   3   2   1]
[  63   15  12  10   4   1]
[  64   12  12   9   2   1]
[  65   12   8   2   1   1]
[  66   12   9   4   1   1]
[  67   21   6   5   2   1]
[  68   15  15  12   5   1]
[  69   15  14   3   2   1]
[  70   12   8   8   1   1]
[  71   18  12   3   2   1]
[  72   48  20  16   3   3]
[  73   22   6   2   2   1]
[  74   16  16  12   3   1]
[  75   15  10   3   1   1]
[  76   24   3   2   2   1]
[  77   15  10   6   1   1]
[  78   18  15   9   2   1]
[  79   24  10   3   2   1]
[  80   18   9   3   1   1]
[  81   24  16   4   3   1]
[  82   14  14   5   1   1]
[  83   18   8   8   1   1]
[  84   22   6   3   1   1]
[  85   20   9   5   1   1]
[  86   18  12   4   1   1]
[  87   18   6   1   1   1]
[  88   30  10   5   2   1]
[  89   30   9   3   2   1]
[  90   21  21  14   3   1]
[  91   20   5   1   1   1]
[  92   20  12   6   1   1]
[  93   42   7   6   6   1]
[  94   30  20  11   4   1]
[  95   18   9   1   1   1]
[  96   24   8   2   1   1]
[  97   33  15   3   3   1]
[  98   20  15   3   1   1]
[  99   30  20  12   3   1]
[ 100   18  15  10   1   1]
[ 101   39   6   4   2   1]
[ 102   30  18   9   2   1]
[ 103   58  15  10   2   2]
[ 104   40  15   5   3   1]
[ 105   24  12   2   1   1]
[ 106   38  12   4   2   1]
[ 107   42  14  10   3   1]
[ 108   30   6   2   1   1]
[ 109   24  16   6   1   1]
[ 110   42   7   3   2   1]
[ 111   30  10   3   1   1]
[ 112   42  12   7   2   1]
[ 113   30  26   6   2   1]
[ 114   55  30  10   2   2]
[ 115   28   4   1   1   1]
[ 116   42  20   4   3   1]
[ 117   42   7   2   2   1]
[ 118   45   9   3   2   1]
[ 119   27  15  10   1   1]
[ 120   48  16   4   3   1]
[ 121   36  24   9   2   1]
[ 122   21  21  12   1   1]
[ 123   36   8   8   1   1]
[ 124   40  24  17   3   1]
[ 125   60   5   5   4   1]
[ 126   24  24   4   1   1]
[ 127   35  28   4   2   1]
[ 128   42  21  11   2   1]
[ 129   45  27   5   3   1]
[ 130   42  36   7   4   1]
[ 131   24  24  10   1   1]
[ 132   42   6   6   1   1]
[ 133   40  10   4   1   1]
[ 134   66  11   6   4   1]
[ 135   42   7   3   1   1]
[ 136   44   5   4   1   1]
[ 137   50  30  15   4   1]
[ 138   38  38  15   3   1]
[ 139   70  18   9   7   1]
[ 140   42   3   2   1   1]
[ 141   60  17   4   3   1]
[ 142   70  15  14   5   1]
[ 143   51  12   2   2   1]
[ 144   70   5   4   4   1]
[ 145   45  30   9   2   1]
[ 146   63   9   9   2   1]
[ 147   45   5   2   1   1]
[ 148   48   8   6   1   1]
[ 149   45  36  29   5   1]
[ 150   60  15  12   2   1]
[ 151   63  36  14  12   1]
[ 152   60  21  20   3   1]
[ 153   45  30   3   2   1]
[ 154   40  20   8   1   1]
[ 155   54  27   6   2   1]
[ 156   30  30   3   1   1]
[ 157   60  35   5   4   1]
[ 158   70   6   5   2   1]
[ 159   42  21   5   1   1]
[ 160   56  28  24   3   1]
[ 161   72   9   8   2   1]
[ 162   48  16   6   1   1]
[ 163   70  13   5   2   1]
[ 164   72  12   9   2   1]
[ 165   72   9   4   2   1]
[ 166   77  33  14   7   1]
[ 167   42  30  30   2   1]
[ 168   66   6   2   2   1]
[ 169   48  16  12   1   1]
[ 170   36  24  18   1   1]
[ 171   84  12   5   3   1]
[ 172   56  11   8   1   1]
[ 173   70  20  12   2   1]
[ 174   60   5   5   1   1]
[ 175   78  26   9   3   1]
[ 176   88  33   3   2   2]
[ 177   54  15  10   1   1]
[ 178   36  28  18   1   1]
[ 179   84  35  14   6   1]
[ 180   42  28  12   1   1]
[ 181   35  34  14   1   1]
[ 182   70  35  35   6   1]
[ 183   51  20  12   1   1]
[ 184   72  36  16   3   1]
[ 185   84  12   6   2   1]
[ 186   72  28   9   2   1]
[ 187   60  15   7   1   1]
[ 188   84  15  10   2   1]
[ 189   60  15   4   1   1]
[ 190   58  18   9   1   1]
[ 191   92  15   4   3   1]
[ 192   42  26  21   1   1]
[ 193   39  35  15   1   1]
[ 194   42  18   1   1   1]
[ 195   56  42   3   2   1]
[ 196   80  48   6   5   1]
[ 197  116  12  10   6   1]
[ 198   70   7   5   1   1]
[ 199   65  14  10   1   1]
[ 200   75  50  20   4   1]
[ 201   63  42  18   2   1]
[ 202  120  16  15   8   1]
[ 203  120  10   8   6   1]
[ 204   60  20   3   1   1]
[ 205  112  16   7   4   1]
[ 206   60  44   3   2   1]
[ 207   75  50   6   3   1]
[ 208   70  14   5   1   1]
[ 209   84  60  14   6   1]
[ 210   42  35  21   1   1]
[ 211  102  24   6   3   1]
[ 212   77  70  11   6   1]
[ 213   78  39  10   2   1]
[ 214   90  60  20   9   1]
[ 215   56  24   2   1   1]
[ 216   85  40  24   3   1]
[ 217   51  30   2   1   1]
[ 218   90  27  15   2   1]
[ 219  120  21  21   5   1]
[ 220   88  33   8   2   1]
[ 221   96  32   4   3   1]
[ 222  120  10  10   3   1]
[ 223   51  42   7   1   1]
[ 224  132  15  12   5   1]
[ 225  126  36  14  12   1]
[ 226   85  40   8   2   1]
[ 227  110  42   7   5   1]
[ 228   66  55  20   2   1]
[ 229   75  60  35   4   1]
[ 230  110  15   3   3   1]
[ 231   45  45  18   1   1]
[ 232  104  65  12  10   1]
[ 233  108  20   3   3   1]
[ 234   56  42   8   1   1]
[ 235   84  12   7   1   1]
[ 236   84  36   3   2   1]
[ 237  140  24   8   7   1]
[ 238   56  42  12   1   1]
[ 239  110  44   6   4   1]
[ 240  106  42   7   3   1]
[ 241  110  15   4   2   1]
[ 242   75  60  12   2   1]
[ 243   65  10   1   1   1]
[ 244  119   7   7   2   1]
[ 245  114  12   4   2   1]
[ 246   90  10   6   1   1]
[ 247   56  48   6   1   1]
[ 248  102  36  12   2   1]
[ 249   78  60  15   2   1]
[ 250   70  30   3   1   1]
[ 251   56  42  20   1   1]
[ 252   70  28  20   1   1]
[ 253   55  44  20   1   1]
[ 254   78  65  10   2   1]
[ 255   84  21  12   1   1]
[ 256   84  20  14   1   1]
[ 257   72  36  10   1   1]
[ 258   63  18   1   1   1]
[ 259   99  22   2   2   1]
[ 260   90  18  10   1   1]
[ 261   84  63  12   2   1]
[ 262  112  35  10   2   1]
[ 263   77  70  15   2   1]
[ 264  105  42  15   2   1]
[ 265  126  21   9   2   1]
[ 266  120  24   5   2   1]
[ 267   56  56  12   1   1]
[ 268  140  35  21   4   1]
[ 269  130  44  39   6   1]
[ 270  156  13   6   4   1]
[ 271  112  48   4   3   1]
[ 272  150  14   7   3   1]
[ 273  135  15   9   2   1]
[ 274  155  20   6   4   1]
[ 275   84  72   9   2   1]
[ 276  168  24   7   7   1]
[ 277   66  48  16   1   1]
[ 278  336  98  48   5   3]
[ 279  105  70   7   3   1]
[ 280  140  10   7   2   1]
[ 281  138  14   6   2   1]
[ 282  105  42  30   2   1]
[ 283  135  40  36   4   1]
[ 284   96  24   6   1   1]
[ 285  154  42   6   6   1]
[ 286   84  24   2   1   1]
[ 287  108  54  15   2   1]
[ 288  104  60  40   3   1]
[ 289  126  21   3   2   1]
[ 290   63  45  30   1   1]
[ 291  144  18  11   2   1]
[ 292  140  60  42   9   1]
[ 293  140  66   7   6   1]
[ 294   95  30   6   1   1]
[ 295  130  20   3   2   1]
[ 296  102  24   8   1   1]
[ 297  100  25   5   1   1]
[ 298  140  62  10   4   1]
[ 299   96  32   8   1   1]
[ 300   77  44  21   1   1]
[ 301   99  22   3   1   1]
[ 302  165  42   7   5   1]
[ 303  140  30   7   2   1]
[ 304  112  16  10   1   1]
[ 305  108  20   5   1   1]
[ 306  120  78   5   4   1]
[ 307  195  30  26  21   1]
[ 308  156  13   3   3   1]
[ 309  153   9   5   2   1]
[ 310   72  40  36   1   1]
[ 311   84  49  12   1   1]
[ 312  168  28   8   3   1]
[ 313  116  15  12   1   1]
[ 314  100  24  24   1   1]
[ 315  120   8   5   1   1]
[ 316  133  14   2   2   1]
[ 317  153  70   9   5   1]
[ 318  180  20  12   3   1]
[ 319  117  18   6   1   1]
[ 320  126  54   6   2   1]
[ 321  114  14   3   1   1]
[ 322   93  30  30   1   1]
[ 323  120  93  30   4   1]
[ 324  100  40   8   1   1]
[ 325  110  11   2   1   1]
[ 326   90  51  10   1   1]
[ 327  210  35  15   9   1]
[ 328  215  24  10   8   1]
[ 329  105  35   5   1   1]
[ 330  153  36  12   2   1]
[ 331  120  64   5   2   1]
[ 332   96  32  30   1   1]
[ 333  102  42   7   1   1]
[ 334  140  50   7   2   1]
[ 335  238   7   6   2   2]
[ 336   84  60  14   1   1]
[ 337  186  21   6   3   1]
[ 338  132 110  12   5   1]
[ 339  168  24  15   2   1]
[ 340  180  36  20   3   1]
[ 341  105  42   6   1   1]
[ 342  210  42  20   7   1]
[ 343   77  70   5   1   1]
[ 344   90  55  18   1   1]
[ 345  120  24   4   1   1]
[ 346  154  66   7   3   1]
[ 347  287 105  12   4   2]
[ 348  156  42   7   2   1]
[ 349  147  42   4   2   1]
[ 350  140  60   7   2   1]
[ 351  119  28   4   1   1]
[ 352   92  20   1   1   1]
[ 353  120  32   6   1   1]
[ 354  210  30   6   5   1]
[ 355  168  32  21   2   1]
[ 356  168  72   7   4   1]
[ 357  240  35  16  14   1]
[ 358  220  22  13   4   1]
[ 359  156  91  21   4   1]
[ 360  130  20   4   1   1]
[ 361  117  39  13   1   1]
[ 362   72  66  36   1   1]
[ 363  210  15   2   2   2]
[ 364  210  21  10   3   1]
[ 365  180  30  12   2   1]
[ 366  124  84   6   2   1]
[ 367  220  25  25   4   1]
[ 368  186  21   7   2   1]
[ 369  112  48   6   1   1]
[ 370  200  40   5   4   1]
[ 371  182  60  14   3   1]
[ 372  195   5   4   3   1]
[ 373  210  56   7   6   1]
[ 374  140 105   5   4   1]
[ 375  189  54  19   3   1]
[ 376   99  66   9   1   1]
[ 377  117  45   5   1   1]
[ 378  220  22   5   4   1]
[ 379  220  23   5   4   1]
[ 380  168 126  16   9   1]
[ 381  189  54   7   3   1]
[ 382  170 119  18   7   1]
[ 383  210  63  35   6   1]
[ 384  126  36   4   1   1]
[ 385  117  52   9   1   1]
[ 386  130  39   9   1   1]
[ 387  210  42  14   3   1]
[ 388  130  95  19   2   1]
[ 389  220  84  14  11   1]
[ 390  168  56   7   2   1]
[ 391  210  35   6   3   1]
[ 392  208  48  13   3   1]
[ 393   91  78  18   1   1]
[ 394  280  21  20  14   1]
[ 395  156  13   9   1   1]
[ 396  150 110  11   3   1]
[ 397  215  70  10   5   1]
[ 398  252  45  30   8   1]
[ 399  195  30   6   2   1]
[ 400  132 110   4   3   1]
[ 401  230  69  48  20   1]
[ 402  231  11   6   3   1]
[ 403  104  65  24   1   1]
[ 404  238  51   7   6   1]
[ 405  154  22  11   1   1]
[ 406  234  26   9   3   1]
[ 407  270  10  10   6   1]
[ 408  120  60  10   1   1]
[ 409  168  98  24   3   1]
[ 410  126  54   7   1   1]
[ 411  168  72   9   2   1]
[ 412  100  70  28   1   1]
[ 413  210  60  20   3   1]
[ 414  160  20  10   1   1]
[ 415  140 100   7   2   1]
[ 416  140 105  12   2   1]
[ 417  231  41  11   3   1]
[ 418  150  24  24   1   1]
[ 419  120  65   8   1   1]
[ 420  238   7   6   3   1]
[ 421  198  44   7   2   1]
[ 422  204  40   8   2   1]
[ 423  170 102  30   3   1]
[ 424  240  16   5   3   1]
[ 425  165  15   5   1   1]
[ 426  102  85  15   1   1]
[ 427  210  70  10   3   1]
[ 428  180 102  20   3   1]
[ 429  126  63   7   1   1]
[ 430  220  30  11   2   1]
[ 431  204  48  17   2   1]
[ 432  198 126  21   6   1]
[ 433  110  82  11   1   1]
[ 434  156  26  24   1   1]
[ 435  255  34   6   4   1]
[ 436  208  48  14   2   1]
[ 437  156  36  13   1   1]
[ 438  231  11   7   2   1]
[ 439  264  68  33   8   1]
[ 440  210  15   3   2   1]
[ 441  174  18   9   1   1]
[ 442  140  56  10   1   1]
[ 443  220 132  21  11   1]
[ 444  130  45  39   1   1]
[ 445  240  48   8   3   1]
[ 446  204  28   3   2   1]
[ 447  150  21   2   1   1]
[ 448  322  21  12  12   1]
[ 449  237  60  15   3   1]
[ 450  238  51   7   3   1]
[ 451  144  48   4   1   1]
[ 452  240  21  16   2   1]
[ 453  168  28  18   1   1]
[ 454  180 110  36   3   1]
[ 455  105  42   1   1   1]
[ 456  132  66  20   1   1]
[ 457  265  35  14   3   1]
[ 458  198 165  22  10   1]
[ 459  176  16   4   1   1]
[ 460  240  65  16   3   1]
[ 461  189 168  12   8   1]
[ 462  240  30  15   2   1]
[ 463  198 110  18   3   1]
[ 464  172  84  42   2   1]
[ 465  297  11  11   4   1]
[ 466  210  60   7   2   1]
[ 467  255  45   2   2   2]
[ 468  168  36  18   1   1]
[ 469  240  35  16   2   1]
[ 470  299  39   3   2   2]
[ 471  120  75   3   1   1]
[ 472  330  36  20   9   1]
[ 473  270  18   5   3   1]
[ 474  252  72   7   4   1]
[ 475  231  49  11   2   1]
[ 476  189  18   7   1   1]
[ 477  305  45   9   6   1]
[ 478  182 182  31   7   1]
[ 479  156  57  13   1   1]
[ 480  104  96  32   1   1]
[ 481  224  96  12   3   1]
[ 482  350  30  25  14   1]
[ 483  336  12  12   7   1]
[ 484  240  80   5   4   1]
[ 485  273  65   6   5   1]
[ 486  234  39  36   2   1]
[ 487  280  63  12   4   1]
[ 488  390 260  13   7   2]
[ 489  285  38  15   3   1]
[ 490  144  72   6   1   1]
[ 491  273  84   3   2   2]
[ 492  210 140  14   4   1]
[ 493  195 156   7   5   1]
[ 494  210  84  15   2   1]
[ 495  288  36  24   3   1]
[ 496  240  10   3   2   1]
[ 497  185  30   5   1   1]
[ 498  240  40   5   2   1]
[ 499  348  30  20   7   1]
[ 500  336  32   8   7   1]
[ 501  248  24   4   2   1]
[ 502  260 140  39  15   1]
[ 503  870 210  35  13   3]
[ 504  161 161  42   3   1]
[ 505  140  56   2   1   1]
[ 506  260  78  18   3   1]
[ 507  231  70  11   2   1]
[ 508  285  76  15   4   1]
[ 509  360  50  24  15   1]
[ 510  273  26  13   2   1]
[ 511  264  33   8   2   1]
[ 512  350  60  25  12   1]
[ 513  154  70   5   1   1]
[ 514  280 100  14   5   1]
[ 515  147  84  12   1   1]
[ 516  238 153   9   7   1]
[ 517  255  85  30   3   1]
[ 518  170  30   2   1   1]
[ 519  182 156  36   3   1]
[ 520  385  15  14  14   1]
[ 521  308  60  12   4   1]
[ 522  273  70  30   3   1]
[ 523  260  78  38   3   1]
[ 524  252  54  27   2   1]
[ 525  165  60   4   1   1]
[ 526  312  27   8   3   1]
[ 527  165  66  22   1   1]
[ 528  252  84  45   3   1]
[ 529  272 160  30  17   1]
[ 530  360  45  10   8   1]
[ 531  294  49   7   3   1]
[ 532  260 140   8   7   1]
[ 533  300  75  10   4   1]
[ 534  232  24   2   2   1]
[ 535  224 192  24   7   1]
[ 536  240  80  12   2   1]
[ 537  322 115  30  15   1]
[ 538  420 294  12   7   2]
[ 539  250 110  11   3   1]
[ 540  154  88   8   1   1]
[ 541  231  93  14   2   1]
[ 542  320  40  20   3   1]
[ 543  168  66  28   1   1]
[ 544  462 375  22  14   2]
[ 545  348  45  36   5   1]
[ 546  189  54   7   1   1]
[ 547  165 160  24   2   1]
[ 548  400  24  16   9   1]
[ 549  240 130  16   3   1]
[ 550  222  21  14   1   1]
[ 551  189  54  21   1   1]
[ 552  200 168  28   3   1]
[ 553  360  45   8   6   1]
[ 554  312 126  21   8   1]
[ 555  143  99  26   1   1]
[ 556  295 105   6   6   1]
[ 557  255 170  27   6   1]
[ 558  280 120  15   4   1]
[ 559  231  14  11   1   1]
[ 560  220  30  12   1   1]
[ 561  285  30   5   2   1]
[ 562  204 170  30   3   1]
[ 563  396  36  22   7   1]
[ 564  390  22  10   6   1]
[ 565  225  18   5   1   1]
[ 566  238 102  14   2   1]
[ 567  220   5   4   1   1]
[ 568  286  52  11   2   1]
[ 569  255  45   3   2   1]
[ 570  165  90  18   1   1]
[ 571  376  24  18   4   1]
[ 572  408  15  10   8   1]
[ 573  315  90  10   4   1]
[ 574  180  63  35   1   1]
[ 575  248  88  33   2   1]
[ 576  341  22   4   4   1]
[ 577  200  50  28   1   1]
[ 578  243 216  18   8   1]
[ 579  280 195  30  14   1]
[ 580  312 120  26   5   1]
[ 581  210  30   3   1   1]
[ 582  130 114  39   1   1]
[ 583  240 111  16   2   1]
[ 584  387  20  18   4   1]
[ 585  209  33   3   1   1]
[ 586  164  77  44   1   1]
[ 587  210  38  35   1   1]
[ 588  420  40  21   8   1]
[ 589  216  27   3   1   1]
[ 590  420  30  14   7   1]
[ 591  308  77   7   3   1]
[ 592  252 180  42   5   1]
[ 593  318  35  15   2   1]
[ 594  176  88  22   1   1]
[ 595  238  51   2   2   1]
[ 596  420  70  22  15   1]
[ 597  336  56  36   3   1]
[ 598  348  20   5   3   1]
[ 599  420  32  21   6   1]
[ 600  330 154  30  10   1]
[ 601  210  60   8   1   1]
[ 602  408  80  40  15   1]
[ 603  273  70   5   2   1]
[ 604  440  44  40  35   1]
[ 605  247 114  26   2   1]
[ 606  154  99  42   1   1]
[ 607  455  42  28  20   1]
[ 608  432  54  15  10   1]
[ 609  330  30  22   2   1]
[ 610  378  27  18   3   1]
[ 611  378  27  14   3   1]
[ 612  420  21  12   5   1]
[ 613  240 176  20   3   1]
[ 614  240  36  10   1   1]
[ 615  342  19  14   2   1]
[ 616  468  20  18  13   1]
[ 617  285 105   5   3   1]
[ 618  390 117  20  12   1]
[ 619  440  60  16  11   1]
[ 620  247 114  39   2   1]
[ 621  232  24   3   1   1]
[ 622  248 220  22   5   1]
[ 623  440  44  10   9   1]
[ 624  300  75  12   2   1]
[ 625  396  88  12   7   1]
[ 626  330  45  18   2   1]
[ 627  225 150  27   2   1]
[ 628  132 132  42   1   1]
[ 629  210 140   4   2   1]
[ 630  336  48   4   3   1]
[ 631  335  40  24   2   1]
[ 632  390  12  10   3   1]
[ 633  380  75  15   4   1]
[ 634  231  42  33   1   1]
[ 635  351 156  24   8   1]
[ 636  369  82  36   4   1]
[ 637  180  90  40   1   1]
[ 638  336   7   6   2   1]
[ 639  420  60  10   6   1]
[ 640  280 200  14   5   1]
[ 641  189  99   7   1   1]
[ 642  462  22  12   7   1]
[ 643  231  60  15   1   1]
[ 644  350  42   4   3   1]
[ 645  312  26   3   2   1]
[ 646  234   9   2   1   1]
[ 647  300  75  42   2   1]
[ 648  187 102   6   1   1]
[ 649  225  45   3   1   1]
[ 650  465  12  10   8   1]
[ 651  270 216  12   5   1]
[ 652  180  90  48   1   1]
[ 653  410  22  15   3   1]
[ 654  340  60   4   3   1]
[ 655  245  42   5   1   1]
[ 656  414 115  36  10   1]
[ 657  325 182  10   7   1]
[ 658  266  28   8   1   1]
[ 659  468  65  39  12   1]
[ 660  354  42  14   2   1]
[ 661  258 130  39   2   1]
[ 662  300 200  22   5   1]
[ 663  350  42  30   2   1]
[ 664  264 220   8   5   1]
[ 665  210  78  35   1   1]
[ 666  264 132   8   2   1]
[ 667  282  42   2   2   1]
[ 668  468  45  39   7   1]
[ 669  416  32  16   3   1]
[ 670  385  99  15   4   1]
[ 671  315  90  12   2   1]
[ 672  280 210   8   5   1]
[ 673  380 108  20   4   1]
[ 674  336 176   8   7   1]
[ 675  266 182  42   3   1]
[ 676  209 190  38   2   1]
[ 677  504  30  16   9   1]
[ 678  378  14  10   2   1]
[ 679  330 120   8   3   1]
[ 680  210  84   4   1   1]
[ 681  288 117   8   2   1]
[ 682  392  70  24   3   1]
[ 683  340  60  39   2   1]
[ 684  525  30  30  14   1]
[ 685  280 112   5   2   1]
[ 686  210  84  40   1   1]
[ 687  330 204  20   6   1]
[ 688  504  63  42  30   1]
[ 689  300 200  15   4   1]
[ 690  483  46  24   6   1]
[ 691  294 126  18   2   1]
[ 692  756 140  42   5   2]
[ 693  238  51   3   1   1]
[ 694  360 135  40   4   1]
[ 695  195 120   8   1   1]
[ 696  462  33  28   4   1]
[ 697  420  72   6   5   1]
[ 698  152 152  36   1   1]
[ 699  210  95  35   1   1]
[ 700  418 171  19  18   1]
[ 701  224  96  14   1   1]
[ 702  308 220  33   5   1]
[ 703  225  90  25   1   1]
[ 704  390  26  10   2   1]
[ 705  435  95  15   5   1]
[ 706  468  45   3   2   2]
[ 707  420  60  21   3   1]
[ 708  315  90   5   2   1]
[ 709  255  68  15   1   1]
[ 710  396  36   4   3   1]
[ 711  240  84  10   1   1]
[ 712  518  15  14   7   1]
[ 713  273  21   3   1   1]
[ 714  420  12   5   3   1]
[ 715  270 216  30   3   1]
[ 716  351 135  30   3   1]
[ 717  480  95  24   8   1]
[ 718  280  45   9   1   1]
[ 719  392 168  21   6   1]
[ 720  476  42  21   4   1]
[ 721  440  10   8   3   1]
[ 722  372  60  30   2   1]
[ 723  357 231  12  11   1]
[ 724  306 135  15   2   1]
[ 725  336 105  20   2   1]
[ 726  273 168  24   2   1]
[ 727  285  45  10   1   1]
[ 728  231 210  11   2   1]
[ 729  374 170  11   5   1]
[ 730  350 140   5   4   1]
[ 731  330 110  30   2   1]
[ 732  460  69  15   4   1]
[ 733  561  45  36  17   1]
[ 734  399  42  12   2   1]
[ 735  546  65  39  35   1]
[ 736  222 208  48   2   1]
[ 737  240 210  16   2   1]
[ 738  532  20  10   7   1]
[ 739  259 182  37   2   1]
[ 740  400 225  24  16   1]
[ 741  483  46  12   4   1]
[ 742  170 170  15   1   1]
[ 743  234 104  20   1   1]
[ 744  315  10   9   1   1]
[ 745  370  78  30   2   1]
[ 746  528  80  40  11   1]
[ 747  450  50  36   3   1]
[ 748  312  26  12   1   1]
[ 749  285 228  15   3   1]
[ 750  510  68  15   6   1]
[ 751  280  63  15   1   1]
[ 752  360 280  30  22   1]
[ 753  312  26  20   1   1]
[ 754  217  28   1   1   1]
[ 755  315 126   6   2   1]
[ 756  546  21  10   7   1]
[ 757  259  74  35   1   1]
[ 758  315  28  15   1   1]
[ 759  246 205  41   2   1]
[ 760  246  99  22   1   1]
[ 761  504  87   9   8   1]
[ 762  414 115   5   5   1]
[ 763  325  13  10   1   1]
[ 764  258 258  39   3   1]
[ 765  260  60   3   1   1]
[ 766  280 160   5   2   1]
[ 767  300 280  14   5   1]
[ 768  420 126   9   4   1]
[ 769  299  39   5   1   1]
[ 770  310  30  30   1   1]
[ 771  414 115   6   4   1]
[ 772  602  35   3   3   2]
[ 773  425  75   6   3   1]
[ 774  460  69   6   4   1]
[ 775  245  98   5   1   1]
[ 776  520  56  42   5   1]
[ 777  390  78  10   2   1]
[ 778  572  88  30  24   1]
[ 779  432 189  27   7   1]
[ 780  594  22  22   9   1]
[ 781  495  60  45   4   1]
[ 782  560  16  12   6   1]
[ 783  483  46  13   3   1]
[ 784  216 144  16   1   1]
[ 785  336 144  21   2   1]
[ 786  352 176  44   3   1]
[ 787  468  55  45   3   1]
[ 788  435 140  29   4   1]
[ 789  420 210  22   7   1]
[ 790  374 264  33   8   1]
[ 791  340 204  17   3   1]
[ 792  420 180   9   6   1]
[ 793  323  38  11   1   1]
[ 794  320  32  30   1   1]
[ 795  451  33   4   3   1]
[ 796  476 102  14   4   1]
[ 797  360 290  36   9   1]
[ 798  330  33  15   1   1]
[ 799  330 220   6   4   1]
[ 800  312   8   3   1   1]
[ 801  299  51  39   1   1]
[ 802  520  84  14   5   1]
[ 803  216 147  27   1   1]
[ 804  612  66  36  33   1]
[ 805  312  52  24   1   1]
[ 806  342  19   9   1   1]
[ 807  336 210  10   3   1]
[ 808  420  70  12   2   1]
[ 809  693 198  28   3   2]
[ 810  494  13   2   2   2]
[ 811  588  98  42  27   1]
[ 812  390 260  13   8   1]
[ 813  200 150  48   1   1]
[ 814  560  70  35   6   1]
[ 815  504  88  33   4   1]
[ 816  429 132   6   4   1]
[ 817  225 150  10   1   1]
[ 818  300  56  42   1   1]
[ 819  325  50  13   1   1]
[ 820  342  19   6   1   1]
[ 821  560  77  16   6   1]
[ 822  330  44  20   1   1]
[ 823  570  51  19   5   1]
[ 824 1470  31   7   6   5]
[ 825  504   9   8   3   1]
[ 826  342  33  19   1   1]
[ 827  325 182  15   2   1]
[ 828  420 280  21  14   1]
[ 829  310  65  26   1   1]
[ 830  416  96   4   3   1]
[ 831  561  44   3   2   2]
[ 832  570  24   8   5   1]
[ 833  228 156  13   1   1]
[ 834  390 117  30   2   1]
[ 835  555 120  19   8   1]
[ 836  240 144  10   1   1]
[ 837  510  63  18   3   1]
[ 838  408 168  15   3   1]
[ 839  312  68  26   1   1]
[ 840  540  90  12   5   1]
[ 841  459  27   3   3   1]
[ 842  975 275  26   9   2]
[ 843  468  39  30   2   1]
[ 844  630  63  14  12   1]
[ 845  630  20  18   7   1]
[ 846  400 112  10   2   1]
[ 847  400 112  35   2   1]
[ 848  480  32  15   2   1]
[ 849  680  30  30  24   1]
[ 850  513 135  30   5   1]
[ 851  320 192  40   2   1]
[ 852  332  45  36   1   1]
[ 853 1170  63  13  12   2]
[ 854  315 210  21   2   1]
[ 855  322  69  21   1   1]
[ 856  299  92  23   1   1]
[ 857  474  70   5   3   1]
[ 858  582  70  21   5   1]
[ 859  564 120  12   8   1]
[ 860  420 315  20  18   1]
[ 861  330 285   6   5   1]
[ 862  408 110  40   2   1]
[ 863  465  45  45   2   1]
[ 864  525 100  42   4   1]
[ 865  368 161  28   2   1]
[ 866  385 252  18   4   1]
[ 867  504 240  36  35   1]
[ 868  675  54  30  15   1]
[ 869  483  46  21   2   1]
[ 870  546 130  42   6   1]
[ 871  483  46  14   2   1]
[ 872  420 168  48   3   1]
[ 873  414 115  35   2   1]
[ 874  360  40  12   1   1]
[ 875  336  56   6   1   1]
[ 876  384 120   5   2   1]
[ 877  399 133  35   2   1]
[ 878  462  56   7   2   1]
[ 879  495 195  18   6   1]
[ 880  270 120  40   1   1]
[ 881  483 224  21   7   1]
[ 882  345 184   8   2   1]
[ 883  380 266  14   4   1]
[ 884  600  42  25   4   1]
[ 885  390 208  48   3   1]
[ 886  456 152  34   3   1]
[ 887 1320 336  11  10   3]
[ 888  630  45  11   6   1]
[ 889  325  85  13   1   1]
[ 890  480  48  45   2   1]
[ 891  476  68  14   2   1]
[ 892  340  60   6   1   1]
[ 893  300  75   3   1   1]
[ 894  392 294  28   5   1]
[ 895  595 140  20   9   1]
[ 896  462 198  28   4   1]
[ 897  231  63   1   1   1]
[ 898  680  51  24   9   1]
[ 899  390 156  36   2   1]
[ 900  492 170  30   4   1]
[ 901  210 195  35   1   1]
[ 902  690  70  30  14   1]
[ 903  330 154   3   2   1]
[ 904  612  45  18   4   1]
[ 905  475 150  17   3   1]
[ 906  585  72  40   4   1]
[ 907  312  96   6   1   1]
[ 908  624 112  35   8   1]
[ 909 1008 112   8   3   3]
[ 910  506  23   7   2   1]
[ 911  384 240  12   3   1]
[ 912  930  69  31   3   2]
[ 913  360  60  18   1   1]
[ 914  693  78  33  14   1]
[ 915  294 105  49   1   1]
[ 916  525 150  20   4   1]
[ 917  320  96  30   1   1]
[ 918  440 110   8   2   1]
[ 919  455 220  35   4   1]
[ 920  520 156   8   5   1]
[ 921  495  66  30   2   1]
[ 922 1185 395  42  35   2]
[ 923  320 105  21   1   1]
[ 924  405 324  20   6   1]
[ 925  448 192  28   3   1]
[ 926  369  44  36   1   1]
[ 927  726  66  33  21   1]
[ 928  455 280  10   8   1]
[ 929  570  70  21   3   1]
[ 930  539 147   3   2   2]
[ 931  390 180  12   2   1]
[ 932  342  76  36   1   1]
[ 933  975 525  39  14   2]
[ 934  336  96  14   1   1]
[ 935  360 216  15   2   1]
[ 936  252 180  21   1   1]
[ 937  315 117  21   1   1]
[ 938  540 108  18   3   1]
[ 939  460 315  20   9   1]
[ 940  588  98  14   4   1]
[ 941  468 102  39   2   1]
[ 942  522  45  10   2   1]
[ 943  312 120  26   1   1]
[ 944  666 148  37  36   1]
[ 945  264 168  28   1   1]
[ 946  483  84  46   2   1]
[ 947  609 210  29  21   1]
[ 948  459 270  30   5   1]
[ 949  630 140  40   8   1]
[ 950  645  68  12   5   1]
[ 951  693  44  11   7   1]
[ 952 1365 156  30  15   2]
[ 953  612 119  28   5   1]
[ 954  372  60  31   1   1]
[ 955  570  90  20   3   1]
[ 956  735  90  28  28   1]
[ 957  630  18   7   4   1]
[ 958  588 168  21   6   1]
[ 959  632  72   3   2   2]
[ 960  468 117  36   2   1]
[ 961  525 231  11   7   1]
[ 962  700  50  20   6   1]
[ 963  696  72  36   7   1]
[ 964  500 210  35   4   1]
[ 965  675  54  15   5   1]
[ 966  357 238  14   2   1]
[ 967  620 180  27   9   1]
[ 968  336 112  12   1   1]
[ 969  275 165  33   1   1]
[ 970  645  30   8   4   1]
[ 971  690  63  45   6   1]
[ 972  520 325  30  24   1]
[ 973  368  80  10   1   1]
[ 974  640 110  11   6   1]
[ 975  504  84   9   2   1]
[ 976  620  21   5   4   1]
[ 977  720  63  48   8   1]
[ 978  469 315  10   9   1]
[ 979  600 150  36   5   1]
[ 980  429 176  16   2   1]
[ 981  294 138  49   1   1]
[ 982  455 260   7   5   1]
[ 983  468 208  40   3   1]
[ 984  570  30   4   3   1]
[ 985 1035 315  23   3   3]
[ 986  546  49  39   2   1]
[ 987  600  80  16   3   1]
[ 988  344 258  40   2   1]
[ 989  273 182  26   1   1]
[ 990  630  45  14   3   1]
[ 991  418 187  38   2   1]
[ 992  600 150  12   5   1]
[ 993  624  52  40   3   1]
[ 994  322 138  21   1   1]
[ 995  572  33  16   2   1]
[ 996  308  84   2   1   1]
[ 997  660  68  15   4   1]
[ 998  630  56  30   3   1]
[ 999  598  78  10   3   1]

26<=n<10000
1=w<=5
y=z and w=s

[    n      x     y     z   w   s]

[    27     2     1     1   1   1]
[    28     2     2     2   1   1]
[    30     3     2     2   1   1]
[    33     3     3     3   1   1]
[    34     4     1     1   1   1]
[    35     4     3     3   1   1]
[    38     6     2     2   1   1]
[    41     3     5     5   1   1]
[    50     6     6     6   1   1]
[    52    10     4     4   1   1]
[    55    12     3     3   1   1]
[    58     6     8     8   1   1]
[    70    12     8     8   1   1]
[    82     5    14    14   1   1]
[    83    18     8     8   1   1]
[   122    12    21    21   1   1]
[   123    36     8     8   1   1]
[   126    36     9     9   1   1]
[   131    10    24    24   1   1]
[   132    42     6     6   1   1]
[   156    42    14    14   1   1]
[   174    60     5     5   1   1]
[   231    60    24    24   1   1]
[   267    12    56    56   1   1]
[   314   100    24    24   1   1]
[   322    93    30    30   1   1]
[   418   150    24    24   1   1]
[   425   115    45    45   1   1]
[   628    42   132   132   1   1]
[   698    36   152   152   1   1]
[   742    15   170   170   1   1]
[   755    60   155   155   1   1]
[   770   310    30    30   1   1]
[   775   266    56    56   1   1]
[  1084    39   246   246   1   1]
[  1340    42   308   308   1   1]
[  1380   418   132   132   1   1]
[  1441   403   155   155   1   1]
[  1882   828    45    45   1   1]
[  1918   646   152   152   1   1]
[  2337   817   171   171   1   1]
[  2766   187   594   594   1   1]
[  2771   780   299   299   1   1]
[  3130    60   744   744   1   1]
[  4122  1558   246   246   1   1]
[  4588  1938   170   170   1   1]
[  5033   403  1053  1053   1   1]
[  5643   266  1273  1273   1   1]
[  5788  2266   308   308   1   1]
[  6172  1890   594   594   1   1]
[  9140    99  2222  2222   1   1]
[  9735  2754  1053  1053   1   1]
[    26     3     3     3   2   2]
[    28     5     3     3   2   2]
[    36    10     5     5   2   2]
[    52    21     7     7   2   2]
[    68    30     3     3   2   2]
[    70    35     5     5   2   2]
[    98    51    15    15   2   2]
[   126    78    13    13   2   2]
[   175    77    42    42   2   2]
[   188    21    75    75   2   2]
[   228   161    21    21   2   2]
[   270   198     9     9   2   2]
[   442   275    75    75   2   2]
[   606    35   273   273   2   2]
[   628   374   119   119   2   2]
[   892   770    33    33   2   2]
[  1084   693   187   187   2   2]
[  1340  1034   141   141   2   2]
[  1380   186   589   589   2   2]
[  2315   105  1090  1090   2   2]
[  2338   275  1023  1023   2   2]
[  2716  2145   273   273   2   2]
[  3130  2666   215   215   2   2]
[  5339  3621   850   850   2   2]
[  5654  5270   153   153   2   2]
[  5876  3813  1023  1023   2   2]
[    34    12     8     8   3   3]
[    41    21     7     7   3   3]
[    58    39    10    10   3   3]
[    62    44     8     8   3   3]
[    82    66     8     8   3   3]
[    87    70     7     7   3   3]
[   108    14    60    60   3   3]
[   140    70    60    60   3   3]
[   267   300    25    25   3   3]
[   345   399    35    35   3   3]
[   371   430    40    40   3   3]
[   418    44   275   275   3   3]
[   551   372   217   217   3   3]
[   742   930    62    62   3   3]
[  1003  1254    19    19   3   3]
[  1036   390   572   572   3   3]
[  1091    70   760   760   3   3]
[  2145  2829   161   161   3   3]
[  3422  4674    38    38   3   3]
[  4267  6006    88    88   3   3]
[  5017  3515  1995  1995   3   3]
[  5151  6726   472   472   3   3]
[  5371  5390  1320  1320   3   3]
[  6567  8284   760   760   3   3]
[  8652   374  6270  6270   3   3]
[  9295 13350   175   175   3   3]
[    28    10     5     5   4   4]
[    42    15    20    20   4   4]
[   156   210    21    21   4   4]
[   370    91   308   308   4   4]
[   388   154   297   297   4   4]
[   459   161   364   364   4   4]
[   468   714    17    17   4   4]
[   551   475   300   300   4   4]
[   566   265   420   420   4   4]
[  2626   111  2516  2516   4   4]
[  2750  2457  1508  1508   4   4]
[  2831  1323  2156  2156   4   4]
[  4083   133  3948  3948   4   4]
[  7359  7881  3404  3404   4   4]
[    27    10     8     8   5   5]
[    33    21     7     7   5   5]
[    41    35     9     9   5   5]
[    42    33     6     6   5   5]
[    58    44    33    33   5   5]
[    59    60     7     7   5   5]
[    82   110    22    22   5   5]
[    98   110     6     6   5   5]
[   122   126    72    72   5   5]
[   123    26   117   117   5   5]
[   131   210    21    21   5   5]
[   132    35   126   126   5   5]
[   174    60   168   168   5   5]
[   187   260     8     8   5   5]
[   267   434    93    93   5   5]
[   270   348   145   145   5   5]
[   290   260   216   216   5   5]
[   299   572    39    39   5   5]
[   330   510   136   136   5   5]
[   345   585    13    13   5   5]
[   388   459   238   238   5   5]
[   468   594   270   270   5   5]
[   470   860   129   129   5   5]
[   606  1260    72    72   5   5]
[   644   437   570   570   5   5]
[   866  1860    88    88   5   5]
[   867  1092   520   520   5   5]
[  2337   549  2623  2623   5   5]
[  3130  7260   121   121   5   5]
[  3300  7410    52    52   5   5]
[  4331   348  5191  5191   5   5]
[  4403 10010   429   429   5   5]
[  4814  4862  3570  3570   5   5]
[  5017   341  6045  6045   5   5]
[  5643  9916  2072  2072   5   5]
[  7843  1260  9144  9144   5   5]
[  8308 19610   106   106   5   5]


4.References

[1]: A. Bremner, R.K. Guy and R.J. Nowakowski, Which integers are representable as the product
     of the sum of three integers with the sum of their reciprocals?, Math. Comp. 61 (1993)

[2]: Yong Zhang, integers representable as the product of the sum of four integers with the sum of their reciprocals,
               https://arxiv.org/abs/1608.03382
[3]: Andrew Bremner and Tho Nguyen Xuan, The equation (w+x+y+z)(1/w+1/x+1/y+1/z)=n, International Journal of Number Theory
     Vol. 14, No. 5 (2018)



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