1.Introduction

We treat diophantine equation (x+y+z+w)(1/x+1/y+1/z+1/w)=n. 
Andrew Bremner and Tho Nguyen Xuan[1] 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).
They pointed out there are two unknown solutions n=10000,15376 when n is of the form 4m^2 where m=2 (mod 4) with n<20000.
In this way there is not always a solution when n is of the form 4m^2 where m=2 (mod 4).
We extended the range up to n=1000000.
As a result, there are unknown solutions below.
[n,m] = [24336, 78], [51984, 114], [63504, 126], [126736, 178], [150544, 194], [169744, 206],
        [176400, 210], [197136, 222], [204304, 226], [226576, 238], [234256, 242], [258064, 254],
        [266256, 258], [274576, 262], [374544, 306], [404496, 318], [414736, 322], [435600, 330],
        [446224, 334], [512656, 358], [535824, 366], [583696, 382], [595984, 386], [633616, 398],
        [646416, 402], [698896, 418], [712336, 422], [725904, 426], [753424, 434], [795664, 446], 
        [810000, 450], [839056, 458], [853776, 462], [913936, 478], [929296, 482], [944784, 486], 
        [992016, 498]
        
Similarly, there is not always a solution when n is of the form 4m^2+4 where m=2 (mod 4).
We extended the range up to n=1000000.
As a result, there are unknown solutions below.
[n,m] = [150548, 194], [156820, 198], [169748, 206], [183188, 214], [197140, 222], [204308, 226],
        [234260, 242], [250004, 250], [258068, 254], [300308, 274], [327188, 286], [355220, 298], 
        [384404, 310], [394388, 314], [425108, 326], [446228, 334], [456980, 338], [478868, 346], 
        [490004, 350], [524180, 362], [559508, 374], [583700, 382], [608404, 390], [646420, 402], 
        [659348, 406], [685588, 414], [712340, 422], [753428, 434], [781460, 442], [795668, 446], 
        [824468, 454], [839060, 458], [868628, 466], [898708, 474], [913940, 478], [929300, 482], 
        [944788, 486], [960404, 490], [976148, 494]
        
2.Results

n=4m^2 where m=2 (mod 4)

[   n     m          x            y           z             w  ]
[ 19600   70      50134733       3053       250217        17457] 
[ 21904   74       3207139       3439          403          247] 
[ 26896   82      32929325       1525        24013         8437] 
[ 29584   86       2977975        143         1323          459] 
[ 32400   90       9552345        437         2145         1573] 
[ 35344   94      29897985      43745   1484870185     26463033] 
[ 38416   98    3806783565     101285     15144909     12769237] 
[ 41616  102        207405        117      3654937      1003145] 
[ 44944  106       4976685       1617          333          185] 
[ 48400  110        732513       1897          153           17] 
[ 55696  118 2477484853217   61614485 847378280625 104397004173] 
[ 59536  122     520655421       8905      3789885      1680305] 
[ 67600  130     167276125       2573      5080125       415245] 
[ 71824  134       9030455       1491     57419475     40613775] 
[ 76176  138    9820801791     141491      3748743      2396995] 
[ 80656  142     478387195      34587   2094307083    216226395] 
[ 85264  146     318699017       4505       106145        27761] 
[ 90000  150    1128951757      13593     77767917     15324433] 
[ 94864  154      20612099        263        68847         1299] 
[ 99856  158    8955490215    2073679 114724107855  83330790615] 
[104976  162    2896391085     115045     12631485        36613] 
[110224  166     505632015       4807      3076227       121011] 
[115600  170     329602595     262655  23672412247   6334870883] 
[121104  174      38730303       1343       122247          423] 
[132496  182         66339        115     14657279       483207] 
[138384  186      10960529         85       663789         7665] 
[144400  190    1066779325       7469      2894925      1936605] 
[156816  198      74883523      77443   6460372503   5596108623] 
[163216  202      72962001        601         5781         2501] 
[183184  214  247823435057    1368753    907491365    247497645] 
[190096  218   31670377375     172007     17584875      7882875] 
[211600  230          2329          5       861045       192321] 
[219024  234       3720475         47      4466363      2106775] 
[242064  246     522096367       2291     26990271      5185323] 
[250000  250  128880972129     523809    244921061     44985501] 
[283024  266   12963512595      50167      3835975       613275] 
[291600  270      29571815        103       316519        28535] 
[300304  274      34192015        119        18183         3135] 
[309136  278     559853791       1963     41200835      5531071] 
[318096  282    2822248585      10101       299065        96621] 
[327184  286   19519227035     471163      2791395        70035] 
[336400  290     506528685       1581        69589        58565] 
[345744  294   20790229713      93641     17405365       170085] 
[355216  298        319495          7      2078475        88275] 
[364816  302   17840829435      55419      1350195       602035] 
[384400  310      50872891        195         1887          527] 
[394384  314   12483514571      34827   1202036615     13288847] 
[425104  326   76516883391     188679    494763971      4595331] 
[456976  338   78991646525     208725      8495165      1141973] 
[467856  342     218767395     272447 118621983723   8462009215] 
[478864  346    6333810725      15345       408525       125281] 
[490000  350     669711917       1413       426933        47437] 
[501264  354    2682061565       5757       486421        90117] 
[524176  362  132904093875     257375     45860339     28903575] 
[547600  370     417011551       4015       694991          943] 
[559504  374      24663093         45         7137         3185] 
[571536  378  540484001707    1212751  77160499955  75461553167] 
[608400  390    1194453943     386307 204601107023  29153687907] 
[620944  394     647668017       1053       759425       156325] 
[659344  406     521204581        801       420453        76857] 
[672400  410    5546203839       8359      5797467       776299]
[685584  414       2044419          3         5963         1691] 
[739600  430    3364590671       4615      2360943       395271] 
[767376  438      69381375         91        54327        24875] 
[781456  442    2381721475       3255       814275        51183] 
[824464  454     988042055       1235       150855        55803] 
[868624  466     535579451        627      2481435        52915] 
[883600  470    2099097143       2387      4343255      2661995] 
[898704  474      49223097      14165   9233815041   3443355685] 
[960400  490      89506359         95      1539555        89091] 
[976144  494     835235225        913       197025        14697] 



n=4m^2+4 where m=2 (mod 4)

[   n     m           x            y           z             w  ]
[ 21908   74         213675         11        23779         1185] 
[ 24340   78        2139093         93         9911         2173] 
[ 26900   82          34083        115      1799865      1248555] 
[ 29588   86         718011       1905     46001305      9481185] 
[ 32404   90       22962249        819        16665         7777] 
[ 35348   94        7210609      20267    683950449     22547817] 
[ 38420   98        9158357        247        47397        10101] 
[ 41620  102         283613          7          855          495] 
[ 44948  106       26064181        879     11941215      1475255] 
[ 48404  110       41344765        871       304381       231693] 
[ 51988  114      987399375      56357       363125        31125] 
[ 55700  118       20280609        371       157563       104427] 
[ 59540  122       42815703       1183     22558029      5038735] 
[ 63508  126       30604365        743        18381         1485] 
[ 67604  130      887147461      13359      5957749      5957749] 
[ 71828  134       68356725        973       928725       309575] 
[ 76180  138        1218581        807     54846141      5360901] 
[ 80660  142      339574235       4521     21555443      2876315] 
[ 85268  146      406304721       4859      3356825       572945] 
[ 90004  150      211389675       2387      1709619       792825] 
[ 94868  154       15769009        195      2523105       185913] 
[ 99860  158         127075         17      1550913        17745] 
[104980  162       10434963        105         7945         2497] 
[110228  166       52313745        489       690305        31339] 
[115604  170      235035757      12117       244629         2471] 
[121108  174        3893747         33         4745         1865] 
[126740  178        4277065         35        10179         1131] 
[132500  182       71070993        561      2640627       524569] 
[138388  186         185493          5       260645       245751] 
[144404  190        1741597         13       117303        16653] 
[163220  202      460098925       3575       996669        13653] 
[176404  210     2177548843      12529     19293521      3167593] 
[190100  218       82509381        551     13983829      8239509] 
[211604  230 69147143045305  329746347 164481857001 103905418617] 
[219028  234      520111515       3465      1021955         7651] 
[226580  238      185964075       1067     37953675     17819385] 
[242068  246      923900327       5917    298249101    210083085] 
[266260  258     2536959685       9879      1052181       367965] 
[274580  262    39779811183     151525     35230503      3735039] 
[283028  266      324303317       1157       355511       251781] 
[291604  270    11469123315      39731     66739185     20510091] 
[309140  278     1296528035       5115       192003        26537] 
[318100  282   180119988113     595273    221717171     12605993] 
[336404  290 46223302644099  138936721  91319850171  19261698291] 
[345748  294       71290989        215      2408175        26271] 
[364820  302      509805291       4667   1124662635     68002857] 
[374548  306     5281491915      14299     51948267      4790889] 
[404500  318        2007445          5         2261         1463] 
[414740  322     1066976559       2735       586245        47311] 
[435604  330      640291465       1555        82419        40051] 
[467860  342      520370165       1141        95029        84535] 
[501268  354        4172315       1633           35           11] 
[512660  358     1650411345       3243      1935505       809393] 
[535828  366     6406335901      12127     19878967      1169351] 
[547604  370     8831990983      17735      2142855       194805] 
[571540  378       24310609         43        60145         5753] 
[595988  386       72541259        123       500745        44555] 
[620948  394    45576425905      73899    154550961    100794105] 
[633620  398    11430165597      18357    147168069      4417367] 
[672404  410      294933697        441       261035       207417] 
[698900  418     7591299581      10959      2659605      2659605] 
[725908  426         792725        781    520059309     45512775] 
[739604  430     3262363461       4743      1808205        65845] 
[767380  438         735745          1        20905        10619] 
[810004  450     2976325209      76129  55131479865   3553930107] 
[853780  462   538014456859     789203  83270843843  52502144945] 
[883604  470      465169149        533      1755351        65013] 
[992020  498        4891511          5         1295          511] 

3.Reference

[1]: 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)