# Problem of the Month (February 2018)

Define the prime signature of a positive integer n to be the exponents to which the prime factors of n are raised. For two prime signatures A and B, what is the smallest number n with prime signature A where the prime signature of n+1 is B?

George Sicherman, Joe DeVincentis, Gordon Atkinson, and Lewis Chen contributed solutions.

n \ n+11211321111431222111111 54132311221211111111
1 23571129none23none593893147711671794192309
2 4none9none49169nonenonenone18492209none57,121none37,24987,784,
138,523,761
24,649214,369
11 6none14265165153935155209none1113911192996293569
3 none8nonenone272197none343none12550,653nonenonenone3,442,951133124,3891,295,029
21 12none2012444153none1889927590924217511242791475185916,925
111 30none105205,378170230none231195494186916,80689032664551547159510,373
4 16none81nonenone130,321nonenonenone1,026,625,6813,418,801nonenonenone2,484,810,622,
489,147,681
none328,385,156,40121,517,662,721
31 402454none152104none135none4591784none16474421624563699171522,329
22 36none225none1444441none69,189,124none864915,129none96,393,124?32491,643,524106,929199,809
211 60none84342260644none3502115819136431244951512412872204405917,289
1111 210none3901330930714none1430115528707314147,008,4422415274353255669910,65940,754
5 nonenone32none2433125nonenonenone161,051371,293nonenonenone16,807?1,350,125,1072,073,071,593
41 112481764912368272805676723142443041,419,85625113,474,2479443824153910,625
32 72none200none1323968none2888675326721,125none3087?171,125344,76333,275830,297
311 27012026468581674594none116044018804185133,923,204,
398,129,542
3320122,824229519,899994421,944
221 180none25221969801044none157551,075338029254,928,488,013,
421,691,156
10,575793,11616,82023,27518,67548,069
2111 420none78024,3889242820none512413,923400490092,476,098494054,924904422,72411,78066,044
11111 2310none39901,225,04219,63510,010none10,37464,51527,39038,570229,345,00615,470336,09030,590119,51085,085378,014
6 nonenone64nonenone729nonenonenone?1,771,561nonenonenonenonenone117,649148,035,889
51 96none160none2528608none483222419522144none1376240,736310451,42410,44939,584
42 400none144none3243969none4624none980158,564none358,420,624?2,424,24971,824257,049866,761
411 240528816670,611,173,776123256062435361520296047849,033,525,
579,302,992
23,2471,723,024699218,51215,34436,464
33 nonenone216nonenone1000none?none274435,937none3375?59,319405,22454,8723,048,625
321 5403605042,588,282,11635002232none66151224617411,304115,856,20016,335219,50015,092111,47527,675156,584
3111 132084015602,048,38283602184none11,960476010,39511,93478,502,725,75029,2951,773,59219,25073,89951,128140,504
222 4356none900none490011,025none101,124none127,44965,025none33,124465,1242,082,2491,249,924915,8493,822,025
2211 2340none12603,869,89273805796none26,775857,475752430,3801,408,514,752,34832,1751,0754,55020,12467,27579,100111,825
21111 4620none54602,685,61811,22013,650none81,62075,07528,86090,80428,629,15098,1244,121,740107,415162,260136,850254,540
111111 43,890none30,03080,062,990124,41053,130none255,2552,390,115184,470393,414714,924,298983,53550,828,946569,4151,103,102447,8102,162,094

n \ n+165142411333213111222221121111111111
1 none223none239none359839none125910,70930,029
2 none?none196249none?94,249none20,187,0492,582,4498,508,889
11 none9514333521575521838992771461939,269
3 nonenonenone148,877none68594913none571,78722,665,18751,895,117
21 63927none135942,216,896,1675393479none32756929229,709
111 none13113991455337493550154355305914,819111,929
4 none?none?none??none?6,216,250,684,0817,538,588,431,201
31 none351783423999991255319none909957,875230,229
22 none53,553,124none651,249none19,740,2491,117,249none28,697,449986,04922,534,009
211 none7351935465592604599717530,27514,65143,757149,225
1111 none41308463721542,87417,83517,76534,59513,29960605255,254
5 nonenonenone229,345,007none1,804,229,351418,195,493none2,476,0991,419,857115,856,201
41 none2672none19,375245,314,37514,49916,119none47,024101,744562,869
32 none41,503none36,125none1,431,125630,125none186,003224,93923,437,829
311 728553520246831185,19211,79919,30427,22431,89958,904399,125
221 none1700?17,77510,64782,12434,83928,89994,275211,9251,824,129
2111 none8415577512,495132,65029,79927,40423,71556,92483,979424,269
11111 none41,055126,735113,295658,502114,85558,69581,795274,274310,1551,774,409
6 nonenonenone186,374,
892,382,561
nonenone279,501,644,934,
771,052,249
none?4,750,104,24110,779,215,329
51 none899158,56311,74435,93614,62426,487none46,899126,9441,349,984
42 none900,471,940,624none15,460,624none544,744,372,62422,024,249none9,865,8811,306,44927,910,089
411 none66,015178,92852649,129,32839,82491,66411,02479,299108,459672,944
33 none?none17,576none42,875238,328none8,365,427274,62555,306,341
321 none48,7351,263,37524,9753,525,688,64741,62412,37538,024105,875248,675982,904
3111 none16,28029,24063,85597,33558,99521,73581,224134,504122,2641,269,729
222 none11,517,153,124none44,796,249none1,762,488,32423,107,249none1,979,64919,421,6494,678,569
2211 none47,775364,815125,775614,12475,140195,975407,043223,244241,6042,804,229
21111 none319,332583,69569,6151,191,015399,399169,575244,035467,3241,241,9542,964,884
111111 none310,3104,145,295899,99024,642,1701,225,0701,565,6551,716,0992,146,1303,139,73011,243,154

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/20/18.