1.Introduction


We treat diophantine equation (x+y+z+w+s+t)(1/x+1/y+1/z+1/w+1/s+1/t)=n. 
In the same way as (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n, there is no positive integer solution for n<36.
We serached the positive integer solutions with 36<=n<10000.
As a result, diophantine equation (x+y+z+w+s+t)(1/x+1/y+1/z+1/w+1/s+1/t)=n has a positive integer solution
for every integer n.
It seems that (x+y+z+w+s+t)(1/x+1/y+1/z+1/w+1/s+1/t)=n has a positive integer solution for any n >=36.
We show the solution table for n<1000.


2.Thorem

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

x = (b+bc+c)a^2+(b^2c+3bc+b+c^2+c+b^2+bc^2)a+bc+bc^2+b^2c
y = abc(a+b+c+1)(n-1)
z = -abc(n-1)
w = bc(a+b+c+1)(n-1)
s = ca(a+b+c+1)(n-1)
t = ab(a+b+c+1)(n-1)

a,b,c,n are arbitrary.

Proof.

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

Multiplying xyzwst to both side, we obtain
(zwst+ywst+yzst+yzwt+yzws)x^2
+(yz^2wt+yw^2st+yz^2ws+zw^2st+y^2zwt+y^2zws+y^2wst+z^2wst
+6yzwst+zwst^2+ywst^2+yzst^2+yzwt^2-yzwst*n+yzw^2t+yzw^2s
+y^2zst+zws^2t+yws^2t+yzs^2t+yzws^2+yz^2st)x
+yzwst^2+y^2zwst+yzw^2st+yzws^2t+yz^2wst=0................................(2)

Let (s,t,w,y,z)=( c(a+b+c+1)au, ab(a+b+c+1)u, bc(a+b+c+1)u, abc(a+b+c+1)u, -abcu ) then (zwst+ywst+yzst+yzwt+yzws) becomes to 0.

Hence we obtain x = (a^2c+a^2b+a^2bc+ac+3abc+ab^2c+ac^2+abc^2+ab^2+ab+bc^2+b^2c+bc)u/(n-1).

Let u=n-1, then x=(b+bc+c)a^2+(b^2c+3bc+b+c^2+c+b^2+bc^2)a+bc+bc^2+b^2c.

Finally, we obtain a parametric solution.

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




3.Results

36<= n <1000

[  n     x    y    z    w    s    t]
[  36    1    1    1    1    1    1] 
[  37    6    6    5    5    4    4]
[  38    4    4    3    3    3    2]
[  39    4    4    3    3    2    2]
[  40    2    2    1    1    1    1]
[  41    8    4    3    3    3    3]
[  42    3    3    3    2    2    1]
[  43    5    5    4    2    2    2]
[  44    3    3    2    2    1    1]
[  45    4    2    2    2    1    1]
[  46   12    5    4    3    3    3]
[  47    8    6    3    3    2    2]
[  48    6    3    2    2    2    1]
[  49    4    3    2    1    1    1]
[  50    4    4    3    2    1    1]
[  51   15    5    5    5    4    2]
[  52    5    1    1    1    1    1]
[  53   17    4    4    3    3    3]
[  54    4    4    1    1    1    1]
[  55    8    6    3    3    3    1]
[  56    6    2    1    1    1    1]
[  57    6    4    4    2    1    1]
[  58    7    2    2    1    1    1]
[  59    6    5    4    3    1    1]
[  60    6    6    2    2    1    1]
[  61   12    5    4    4    4    1]
[  62    9    3    2    2    1    1]
[  63   12    6    3    3    3    1]
[  64   12    4    3    2    2    1]
[  65   14    4    3    3    3    1]
[  66    6    6    3    1    1    1]
[  67   10    4    2    2    1    1]
[  68    9    3    3    1    1    1]
[  69   24    8    5    3    3    2]
[  70    8    4    1    1    1    1]
[  71   26   13    4    3    3    3]
[  72   14    6    3    2    2    1]
[  73   24    5    4    3    2    2]
[  74   12    4    4    2    1    1]
[  75   24    6    3    3    2    2]
[  76   10    5    2    1    1    1]
[  77   12   12    3    3    2    1]
[  78   12    6    2    2    1    1]
[  79   22   12    3    3    2    2]
[  80   13    6    3    2    1    1]
[  81   10   10    4    4    1    1]
[  82   24    8    3    2    2    2]
[  83   15    5    4    4    1    1]
[  84   18    9    3    3    2    1]
[  85    8    8    1    1    1    1]
[  86   12    6    3    1    1    1]
[  87   18    9    4    2    2    1]
[  88   20    5    2    2    2    1]
[  89   24   21    3    3    3    2]
[  90   30   10    3    3    2    2]
[  91   24    8    3    3    3    1]
[  92   45    9    5    4    4    2]
[  93   12   12    3    2    1    1]
[  94   16    8    4    2    1    1]
[  95   12   12    2    2    1    1]
[  96   15   10    3    2    1    1]
[  97   14    7    4    1    1    1]
[  98   20    6    4    3    1    1]
[  99   20    4    2    2    1    1]
[ 100   24    8    3    2    2    1]
[ 101   12    8    1    1    1    1]
[ 102   18    9    3    2    1    1]
[ 103   20    7    4    2    1    1]
[ 104   24    8    2    2    2    1]
[ 105   12   12    3    1    1    1]
[ 106   20   10    4    4    1    1]
[ 107   36   24    4    3    3    2]
[ 108   48    4    4    3    3    2]
[ 109   20    4    3    1    1    1]
[ 110   15   10    2    1    1    1]
[ 111  105   12    6    6    6    5]
[ 112   20   12    3    3    1    1]
[ 113   18   12    2    2    1    1]
[ 114   21   12    4    3    1    1]
[ 115   16   16    4    2    1    1]
[ 116   42    7    3    2    2    2]
[ 117   24    8    3    2    1    1]
[ 118   60    8    3    3    3    3]
[ 119   48   20    4    3    3    2]
[ 120   28   21    4    3    3    1]
[ 121   35   14    4    3    3    1]
[ 122   27   18    3    3    2    1]
[ 123   30    5    4    4    1    1]
[ 124   24    3    2    1    1    1]
[ 125   18   12    3    1    1    1]
[ 126   35   14    4    4    2    1]
[ 127   39    4    3    3    2    1]
[ 128   30    3    3    2    1    1]
[ 129   36    9    4    2    2    1]
[ 130   30   20    4    3    2    1]
[ 131   74   20    6    5    4    2]
[ 132   42    4    4    3    2    1]
[ 133   24    8    3    1    1    1]
[ 134   44   11    4    3    3    1]
[ 135   28   21    6    2    2    1]
[ 136   39   15    4    4    2    1]
[ 137   33    4    3    2    1    1]
[ 138   40   30    3    3    2    2]
[ 139   24   10    3    1    1    1]
[ 140   39    6    2    2    2    1]
[ 141   45   36    5    4    2    2]
[ 142  100   25    5    4    4    4]
[ 143   51    6    4    3    3    1]
[ 144   28    7    4    1    1    1]
[ 145   78   60    5    5    4    4]
[ 146   60   60    7    6    5    2]
[ 147   69   12    3    3    3    2]
[ 148   84   14    4    3    3    3]
[ 149   76   42    4    4    4    3]
[ 150   24    8    1    1    1    1]
[ 151   20   18    4    1    1    1]
[ 152   24   24    4    3    1    1]
[ 153   36   36    3    2    2    2]
[ 154   32   24    3    2    2    1]
[ 155   30   20    4    4    1    1]
[ 156   60   20    3    3    2    2]
[ 157   95    5    4    4    3    3]
[ 158   60    8    4    4    3    1]
[ 159   72   24    4    3    3    2]
[ 160   42    6    3    3    1    1]
[ 161   60   15    3    2    2    2]
[ 162   20   20    2    1    1    1]
[ 163   88   55    8    6    6    2]
[ 164   70   30    4    3    3    2]
[ 165   24   12    1    1    1    1]
[ 166   85   68    5    4    4    4]
[ 167   84   16    4    3    3    2]
[ 168   30    6    1    1    1    1]
[ 169   70    7    5    4    4    1]
[ 170   36    4    2    1    1    1]
[ 171  110   11    4    4    3    3]
[ 172   70    5    3    2    2    2]
[ 173   84   42    5    3    3    3]
[ 174   48   16    3    2    2    1]
[ 175   40   10    2    2    1    1]
[ 176   60   10    4    3    2    1]
[ 177  120   28    7    6    5    2]
[ 178   84   10    4    3    2    2]
[ 179   39    6    4    1    1    1]
[ 180   30   15    2    1    1    1]
[ 181  145   12    5    4    4    4]
[ 182   30   30    5    3    1    1]
[ 183   28   28    3    2    1    1]
[ 184   28   20    5    1    1    1]
[ 185   36   18    2    2    1    1]
[ 186   36   12    3    1    1    1]
[ 187   40    8    4    1    1    1]
[ 188   45    9    2    2    1    1]
[ 189   60   15    3    3    2    1]
[ 190   30   20    4    1    1    1]
[ 191  132   15    6    6    4    2]
[ 192   35   15    3    1    1    1]
[ 193   52    5    4    2    1    1]
[ 194   50   15    4    4    1    1]
[ 195   42    8    3    1    1    1]
[ 196   60   15    4    2    2    1]
[ 197   56   40    3    2    2    2]
[ 198   42    7    2    1    1    1]
[ 199  115   10    4    4    3    2]
[ 200   70    5    3    3    2    1]
[ 201   84   42    5    5    2    2]
[ 202   88    8    5    4    4    1]
[ 203   80   48    4    3    3    2]
[ 204   48   16    4    2    1    1]
[ 205   36   18    3    1    1    1]
[ 206  112   35    4    3    3    3]
[ 207   70    5    4    2    2    1]
[ 208   45    9    3    1    1    1]
[ 209   85   40    3    3    3    2]
[ 210   60   40    4    4    3    1]
[ 211  140   90    7    7    5    3]
[ 212   60   10    5    3    1    1]
[ 213   84   10    4    3    3    1]
[ 214   92   10    4    4    4    1]
[ 215   77   44    6    6    6    1]
[ 216  145   42    5    4    4    3]
[ 217  120    8    4    3    3    2]
[ 218   84   14    4    3    3    1]
[ 219   70   42    5    4    4    1]
[ 220   56    8    2    2    1    1]
[ 221   60   60    3    3    2    2]
[ 222   72   24    6    3    2    1]
[ 223   57   12    3    2    1    1]
[ 224   60   15    4    3    1    1]
[ 225   84   14    3    3    3    1]
[ 226  120   34    8    3    3    2]
[ 227  217    8    7    7    6    3]
[ 228   45    5    1    1    1    1]
[ 229   72   36    5    3    3    1]
[ 230  144   48    4    4    4    3]
[ 231   72    9    2    2    2    1]
[ 232   56   40    3    3    2    1]
[ 233   98   84    4    4    3    3]
[ 234   84   14    4    3    2    1]
[ 235   70    5    4    3    1    1]
[ 236  111   60    4    4    4    2]
[ 237   45   36    4    3    1    1]
[ 238   40   24    3    1    1    1]
[ 239   72   36    5    4    2    1]
[ 240   70   42    6    5    2    1]
[ 241  170   17    6    5    4    2]
[ 242  140   15    5    3    3    2]
[ 243  100   25    4    2    2    2]
[ 244  156   13    4    4    4    2]
[ 245   51   12    2    1    1    1]
[ 246  105   21    5    4    4    1]
[ 247   48   48    3    2    2    1]
[ 248   54   27    5    2    1    1]
[ 249  130   52    4    3    3    3]
[ 250   90   40    5    4    4    1]
[ 251  204   34    5    4    4    4]
[ 252   60    4    3    1    1    1]
[ 253  100   25    5    4    3    1]
[ 254  246   21    6    6    4    4]
[ 255   60   30    4    4    1    1]
[ 256  140    7    5    4    2    2]
[ 257  120  120    5    5    4    3]
[ 258   75   15    4    4    1    1]
[ 259   78    4    4    3    1    1]
[ 260   84   38    4    3    3    1]
[ 261  168    8    4    3    3    3]
[ 262  140  105    5    5    4    3]
[ 263  216   27    8    6    3    3]
[ 264   90   10    3    2    2    1]
[ 265  105   20    6    6    2    1]
[ 266   80   48    5    3    3    1]
[ 267  150  100    5    5    4    3]
[ 268  168   42    4    4    3    3]
[ 269   84   10    5    4    1    1]
[ 270  100   15    4    3    2    1]
[ 271  140  120    7    7    3    3]
[ 272  156   14    4    3    3    2]
[ 273  160   96    5    5    4    3]
[ 274   44   28    2    1    1    1]
[ 275  117   52    3    3    3    2]
[ 276  130   39    3    3    3    2]
[ 277   85   40    4    4    2    1]
[ 278  260  168    8    7    6    6]
[ 279  102   24    3    3    3    1]
[ 280  120    8    5    3    3    1]
[ 281  108   54    6    6    5    1]
[ 282   49   42    3    2    1    1]
[ 283  112   16    5    4    2    1]
[ 284  156   39    4    4    3    2]
[ 285  156   13    3    3    3    2]
[ 286   66   55    3    3    2    1]
[ 287   84    6    4    2    1    1]
[ 288  100   25    4    3    2    1]
[ 289  120   40    3    3    2    2]
[ 290  117   52    4    3    2    2]
[ 291  130   39    4    3    2    2]
[ 292  210   38    6    5    5    2]
[ 293   72   36    5    5    1    1]
[ 294  120   12    5    4    2    1]
[ 295  207    9    4    4    3    3]
[ 296   70   60    4    3    2    1]
[ 297   72    9    4    1    1    1]
[ 298   90   18    6    4    1    1]
[ 299  112   84    4    3    3    2]
[ 300   90   10    3    3    1    1]
[ 301  132  110    5    5    4    2]
[ 302  140   85    4    3    3    3]
[ 303   56   48    3    3    1    1]
[ 304  176   66    4    4    3    3]
[ 305  155   20    4    3    2    2]
[ 306  204   85    5    5    4    3]
[ 307  132   84    7    3    3    2]
[ 308  308   35    6    5    5    4]
[ 309   90   10    4    2    1    1]
[ 310  204   21    4    3    3    3]
[ 311  117   52    5    2    2    2]
[ 312   68   42    4    3    1    1]
[ 313   91   52    7    3    2    1]
[ 314  170   12    8    8    5    1]
[ 315  240  144    6    5    5    5]
[ 316  120   88    4    3    3    2]
[ 317  231   14    8    6    3    2]
[ 318  220   30    4    4    3    3]
[ 319   88   24    4    3    1    1]
[ 320  220   22    5    3    3    3]
[ 321  156   13    5    2    2    2]
[ 322  132   12    3    3    3    1]
[ 323   96   32    3    2    2    1]
[ 324   84   21    3    2    1    1]
[ 325  132   22    4    3    3    1]
[ 326  160   15    8    4    4    1]
[ 327  276   28    7    4    4    3]
[ 328   91    7    2    2    1    1]
[ 329  330   88    6    6    5    5]
[ 330  195   26    5    3    3    2]
[ 331  187  102    7    6    4    2]
[ 332  216   40    5    3    3    3]
[ 333  240   16    4    4    3    3]
[ 334  120   44    5    3    3    1]
[ 335  270   11    5    4    4    3]
[ 336  135   15    6    3    2    1]
[ 337  240   64    5    4    4    3]
[ 338  258   28    4    4    4    3]
[ 339  272   16    8    4    3    3]
[ 340  110   15    2    2    2    1]
[ 341   56   24    1    1    1    1]
[ 342   80   48    2    2    2    1]
[ 343   80   48    6    4    1    1]
[ 344  132   33    4    3    3    1]
[ 345   85   40    6    3    1    1]
[ 346  187   55    4    4    3    2]
[ 347  102   24    4    4    1    1]
[ 348   96   24    3    3    1    1]
[ 349  280   63    5    4    4    4]
[ 350   84   14    4    1    1    1]
[ 351   56   42    3    1    1    1]
[ 352  216   27    4    4    3    2]
[ 353  195   52    4    4    3    2]
[ 354   90   36    4    3    1    1]
[ 355  140   60    3    3    2    2]
[ 356  171   45    5    3    2    2]
[ 357  140  140    5    3    3    3]
[ 358  260   23    5    5    4    2]
[ 359   70   60    5    3    1    1]
[ 360  153    9    2    2    2    2]
[ 361  280   63    5    5    4    3]
[ 362  168   72    4    3    3    2]
[ 363  210   28    5    5    2    2]
[ 364  110   11    5    2    1    1]
[ 365  117   52    4    4    2    1]
[ 366  130   39    4    4    2    1]
[ 367  155   20    4    3    3    1]
[ 368   88   33    3    2    1    1]
[ 369  182   78    4    4    3    2]
[ 370  140   90    8    8    5    1]
[ 371   60   44    3    1    1    1]
[ 372  130   39    5    3    2    1]
[ 373  280  120    7    5    5    3]
[ 374  132  110    5    4    2    2]
[ 375  156   13    4    4    2    1]
[ 376  105   70    7    3    2    1]
[ 377  120   68    8    5    2    1]
[ 378  285   76    5    5    4    3]
[ 379  130   52    6    4    2    1]
[ 380  210   42    5    5    2    2]
[ 381  156   13    5    3    2    1]
[ 382  434   40    7    5    5    5]
[ 383   84   36    2    2    1    1]
[ 384  110   15    3    2    1    1]
[ 385  420   21    6    6    5    4]
[ 386  216   40    5    5    2    2]
[ 387   80   48    3    2    1    1]
[ 388  287  105    5    5    5    3]
[ 389  224   96    7    3    3    3]
[ 390  168   28    4    4    3    1]
[ 391   56   56    4    1    1    1]
[ 392  180   60    5    3    2    2]
[ 393  152    8    6    2    2    1]
[ 394  140  110    7    3    2    2]
[ 395  117   52    6    2    2    1]
[ 396  312   24    4    4    4    3]
[ 397  112   48    3    2    2    1]
[ 398  360   54    5    5    4    4]
[ 399  132   12    4    4    1    1]
[ 400  116   20    5    2    1    1]
[ 401  164   30    4    3    3    1]
[ 402  300   35    5    4    3    3]
[ 403  105   28    3    2    1    1]
[ 404  333   18    6    6    5    2]
[ 405  156   13    6    2    2    1]
[ 406  320   48    5    4    4    3]
[ 407  224  112    7    3    3    3]
[ 408  330  154    5    5    5    5]
[ 409  132  120    5    3    2    2]
[ 410  130   90    5    5    3    1]
[ 411  306   18    5    5    4    2]
[ 412  120   30    5    3    1    1]
[ 413  132   22    5    4    1    1]
[ 414  231  132    4    4    4    3]
[ 415  120   88    5    3    3    1]
[ 416   66   55    6    1    1    1]
[ 417  224   96    7    4    3    2]
[ 418  306  135    5    5    4    4]
[ 419  315   28    8    3    3    3]
[ 420  180   36    3    2    2    2]
[ 421  140   85    4    4    4    1]
[ 422  315  144    7    7    4    3]
[ 423  150   40    4    3    2    1]
[ 424  354   42    6    4    4    3]
[ 425   84   36    3    1    1    1]
[ 426  140   60    7    3    2    1]
[ 427  198   44    3    3    2    2]
[ 428  165   44    4    3    3    1]
[ 429  260   65    4    3    3    3]
[ 430  180    9    5    3    2    1]
[ 431  195  104    5    3    3    2]
[ 432  171   45    5    3    3    1]
[ 433  120   32    3    3    1    1]
[ 434  315   28    6    6    3    2]
[ 435  360   72    6    5    4    3]
[ 436  205   70    4    4    2    2]
[ 437   84   42    4    1    1    1]
[ 438  195   30    3    2    2    2]
[ 439  110   15    4    1    1    1]
[ 440  380   20    6    6    3    3]
[ 441  156   12    6    6    1    1]
[ 442  132    4    3    2    1    1]
[ 443  369   20    9    6    4    2]
[ 444  231   11    3    3    2    2]
[ 445  100   20    2    1    1    1]
[ 446  117   52    5    4    1    1]
[ 447   84   69    4    2    1    1]
[ 448   99   22    2    1    1    1]
[ 449  360   72    5    5    4    3]
[ 450  152    8    6    3    1    1]
[ 451  207   36    4    2    2    2]
[ 452  115   15    5    1    1    1]
[ 453  112   48    3    3    1    1]
[ 454   84   24    1    1    1    1]
[ 455  204   85    3    3    3    2]
[ 456  156   13    5    4    1    1]
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[ 920  630   45    7    3    3    2]
[ 921  440   40    6    6    2    1]
[ 922 1008   90    9    5    4    4]
[ 923  612   34    9    4    2    2]
[ 924  336   48    3    2    2    1]
[ 925  420  168    5    3    2    2]
[ 926  255   45    2    2    1    1]
[ 927  460   69    5    2    2    2]
[ 928  660   30    5    4    3    2]
[ 929  728   72    7    7    3    2]
[ 930  495   45    3    3    2    2]
[ 931  520  104    5    4    2    2]
[ 932  477  252    4    3    3    3]
[ 933 1140   35    6    6    6    4]
[ 934  840  480    8    6    5    5]
[ 935  980   49    7    7    4    3]
[ 936  828  230    9    6    4    3]
[ 937  294   84    8    4    1    1]
[ 938 1170   13    6    6    6    5]
[ 939 1365  168    7    7    7    6]
[ 940  385  154    5    2    2    2]
[ 941  429  176    6    6    6    1]
[ 942  504  144    8    8    7    1]
[ 943  540   60    8    8    4    1]
[ 944  350   42    3    2    2    1]
[ 945  680  476    5    5    5    5]
[ 946  780   65    4    4    4    3]
[ 947  150  140    7    1    1    1]
[ 948 1005   45    9    6    4    3]
[ 949  420   72    5    3    3    1]
[ 950 1260  420   10    7    7    6]
[ 951 1192  120   10   10    6    3]
[ 952  792   72    5    5    3    3]
[ 953  280  132   14   12    1    1]
[ 954  264   24    6    1    1    1]
[ 955  726  462    7    7    4    4]
[ 956  268   60    3    2    1    1]
[ 957  720   48    5    4    4    2]
[ 958  595   70    7    4    2    2]
[ 959 1120   32    8    8    5    3]
[ 960  340   60    3    2    2    1]
[ 961  462  264    7    7    2    2]
[ 962  285   76    4    3    1    1]
[ 963  696  145    5    4    3    3]
[ 964  462  143    7    7    4    1]
[ 965 1080   88    5    5    5    5]
[ 966 1190  255    7    7    6    5]
[ 967  429  176    8    6    4    1]
[ 968  680   72    4    3    3    3]
[ 969  495  352    6    5    4    2]
[ 970  390  180    5    5    4    1]
[ 971  511   56    7    6    3    1]
[ 972  300   42    4    2    1    1]
[ 973  918  135    5    5    4    4]
[ 974 1188  108    9    6    5    4]
[ 975  408    8    3    3    2    1]
[ 976  598   78    5    5    2    2]
[ 977  483   56    4    4    4    1]
[ 978 1035   23    9    6    4    3]
[ 979  341   22    4    3    1    1]
[ 980  315  126    3    3    2    1]
[ 981 1295   12    7    7    6    5]
[ 982  572   33    8    6    4    1]
[ 983  516   70    6    5    4    1]
[ 984 1098  220    9    6    5    4]
[ 985  490  196    5    5    2    2]
[ 986  924   44    7    4    4    3]
[ 987  476  102    3    3    2    2]
[ 988  990   36    6    6    4    3]
[ 989  624   52    4    3    3    2]
[ 990  798  133    6    6    5    2]
[ 991  420   72    5    4    2    1]
[ 992  840   10    8    4    3    3]
[ 993  513  216   10   10   10    1]
[ 994  665   57    4    4    3    2]
[ 995 1512   81    8    7    6    6]
[ 996  390   10    8    6    1    1]
[ 997  312  282    8    4    4    1]
[ 998  600   54    8    6    6    1]
[ 999  660   66    6    3    3    2]


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)