TITLE: All positive integer solutions to $\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$
QUESTION [19 upvotes]: As the title states, how would I go about finding the positive integer solutions of
$$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$$?
Thank you for your help.
Edit: I think I've made some progress. I conjecture that you can expand the set {a, a+1, a*(a+1)+1, a*(a+1)(a(a+1)+1)+1, ...} (i.e. multiply all previous solutions and add 1), where a=2, to size n, and that will always be a solution in the case of n variables. This is far from finding all positive integer solutions, though.
Edit 2: For example {2}, {2,3}, {2,3,7}, {2,3,7,43}, {2,3,7,1807}, {2,3,7,43,1807,3263443} are all solutions in the case where $n=$ size of the solution set respectively.

REPLY [9 votes]: As the link mentioned in the Mathematics Magazine article is inaccessible, I have rerun the code described in the article, and I report the results (122 octets for $n=8$) here.  It appears that the run described in the article missed three of the octets.  I assume that the missing octets are the ones mentioned in the OEIS article found by Zhao Hui Du.
2 3 7 43 1807 3263443  10650056950807 113423713055421844361000443
2 3 7 43 1807 3263443  10652778201539 41691378583707695
2 3 7 43 1807 3263443  10699597306267 2300171639909623
2 3 7 43 1807 3263443  10650057792155 134811739261383753719
2 3 7 43 1807 3263447  2130014000915 22684798220467498090185211
2 3 7 43 1807 3263447  2130014387399 11739058070963394487
2 3 7 43 1807 3263479  288182779055 243811701792623
2 3 7 43 1807 3263483  260604226747 80249212735823
2 3 7 43 1807 3263495  200947673239 67137380077902268343
2 3 7 43 1807 3263495  200949404503 23316080984691959
2 3 7 43 1807 3263531  119666789791 8081907028348841339
2 3 7 43 1807 3263591  71480133827 761302020256877140089595
2 3 7 43 1807 3263779  31834629787 4396910340967
2 3 7 43 1807 3264187  14298637519 152316021000302785506427
2 3 7 43 1807 3316627  203509259 109643149191047
2 3 7 43 1807 3586039  36800447 2550097247
2 3 7 43 1811 655519  389313431 1507818475
2 3 7 43 1811 713899  7813583 2409102303622951
2 3 7 43 1811 793595  3722287 233296531681207
2 3 7 43 1817 298637  279594269 3859101523354821017
2 3 7 43 1819 252731  2134319143 6047845668256680791
2 3 7 43 1823 193667  637617223447 406555723635623909338363
2 3 7 43 1823 193667  637617223459 31273517203328870463055
2 3 7 43 1823 193675  4683210919 754794584867
2 3 7 43 1831 132347  231679879 1197240789041771
2 3 7 43 1891 40379  9444811 55866875
2 3 7 43 1943 25615  456729463 450222796871
2 3 7 43 1951 30571  118463 14484098803019
2 3 7 43 2105 12773  2775277 168100338289
2 3 7 43 2137 16921  37501 49708999789
2 3 7 43 2755 5407  172771 357538828973647
2 3 7 43 2813 5045  692705317 188433744928309
2 3 7 43 3263 4051  2558951 61088439723561979
2 3 7 43 3559 3667  33816127 797040720326433787
2 3 7 47 395 779731  607979652631 369639258012703445569531
2 3 7 47 395 779731  607979655287 139119028839856004123
2 3 7 47 395 779731  607979652683 6974325623477705424647
2 3 7 47 395 779731  607979793451 2624887933109395111
2 3 7 47 395 779731  607979652647 21743485766025360000683
2 3 7 47 395 779731  607979697799 8183472856913555659
2 3 7 47 395 779731  607979653531 410254449012081168631
2 3 7 47 395 779731  607982046587 154405744751990423
2 3 7 47 395 779743  46768385339 1672627310178141725483
2 3 7 47 395 779747  35764242947 12154487527525118239
2 3 7 47 395 779827  6286857907 2158880732959
2 3 7 47 395 779831  6020372531 3660733426607933569531
2 3 7 47 395 781727  305967719 125881309327
2 3 7 47 395 782111  257276179 57278664659
2 3 7 47 395 782287  277442411 1701723083
2 3 7 47 395 782611  211810259 1592773460578079
2 3 7 47 395 816247  17428931 652510750371360683
2 3 7 47 395 1108727  2627707 140495574531059
2 3 7 47 403 19403  15435513367 238255072887400163323
2 3 7 47 403 19403  15435513463 2456237880094942747
2 3 7 47 403 19403  15435513395 8215692183434294399
2 3 7 47 403 19403  15435516179 84697872837562655
2 3 7 47 415 8111  6644612311 44150872756848148411
2 3 7 47 415 8111  6644613463 38292177286592827
2 3 7 47 415 8111  6644612339 1522443894582665279
2 3 7 47 415 8111  6644645747 1320426321921983
2 3 7 47 449 4477  12137 34035763385
2 3 7 47 583 1223  1407479767 1980999293106894523
2 3 7 47 583 1223  2202310039 3899834875
2 3 7 47 583 1223  1468268915 33995520959
2 3 7 47 583 1223  1407479807 48317057302587443
2 3 7 53 209 10589  19651 86321
2 3 7 53 269 817  7301713 48932949591475
2 3 7 53 401 409  351691 397617853
2 3 7 55 179 24323  10057317271 101149630679497570171
2 3 7 55 179 24323  10057317467 513449911932648503
2 3 7 55 179 24323  10057317311 2467064172726591731
2 3 7 55 179 24323  10057325347 12523178395739983
2 3 7 55 179 24323  10057317287 5949978284730273323
2 3 7 55 179 24323  10057320619 30202945461748519
2 3 7 55 179 24323  10057317967 145121431390804003
2 3 7 55 179 24323  10057454579 736667018400959
2 3 7 61 187 485  150809 971259409
2 3 7 61 293 457  551 21709309
2 3 7 65 121 6271  1579937 2869621
2 3 7 67 113 28925  48220169 4074021053
2 3 7 67 113 29153  3712777 45401353
2 3 7 67 113 34477  178945 178344158228021
2 3 7 67 187 283  334651 49836124516795
2 3 7 71 103 65059  1101031 4400294969594807
2 3 11 17 79 301  1049 3696653
2 3 11 17 97 151  444161 317361415625
2 3 11 17 101 149  3109 52495396603
2 3 11 23 31 47059  2214502423 4904020979258368507
2 3 11 23 31 47059  2294166883 63772955407
2 3 11 23 31 47059  2215070383 8636647107907
2 3 11 23 31 47059  2214502831 11990273552017987
2 3 11 23 31 47059  2446798471 23325584587
2 3 11 23 31 47059  2214502475 92528699894575367
2 3 11 23 31 47059  2244604355 165128325167
2 3 11 23 31 47059  2214524099 226233749172527
2 3 11 23 31 47059  2214502427 980804197623275639
2 3 11 23 31 47059  2612824727 14526193019
2 3 11 23 31 47059  2217342227 1729101023519
2 3 11 23 31 47059  2214504467 2398056482005535
2 3 11 23 31 47059  3375982667 6436718855
2 3 11 23 31 47059  2214502687 18505741750517011
2 3 11 23 31 47059  2365012087 34797266971
2 3 11 23 31 47059  2214610807 45248521436443
2 3 11 23 31 47063  442938131 980970939025927675
2 3 11 23 31 47063  447473399 43702604167
2 3 11 23 31 47095  59897203 132743972247361531
2 3 11 23 31 47119  36349891 4619150372467
2 3 11 23 31 47131  30382063 67384091875543675
2 3 11 23 31 47147  24928579 11061526082145911
2 3 11 23 31 47243  12017087 26715920281613179
2 3 11 23 31 47423  6114059 13644326865136507
2 3 11 23 31 47479  5307047 2371471764522551
2 3 11 23 31 47491  5161279 4952592862147
2 3 11 23 31 49759  866923 2029951372029307
2 3 11 23 31 60563  211031 601432790177275
2 3 11 23 31 74963  126415 259118345891
2 3 11 23 31 84527  106159 84453127154999
2 3 11 25 29 787  264841 2542873
2 3 11 25 29 1097  2753 144508961851
2 3 11 31 35 67  369067 1770735487291
2 3 13 25 29 67  2981 11294561851
2 5 7 11 17 157  961 4398619