A238194 -id:A238194 - cp's OEIS frontend

A283146 Prime numbers p whose square divides a number of the form n^n + (-1)^n (n-1)^(n-1), where n is a positive integer.

59, 83, 179, 193, 337, 419, 421, 443, 457, 547, 601, 619, 701, 787, 857, 887, 911, 929, 977, 1039, 1091, 1093, 1109, 1193, 1217, 1223, 1237, 1259, 1289, 1439, 1487, 1489, 1493, 1613, 1637, 1657, 1811, 1847, 1901, 1993, 1997, 2003, 2087, 2089, 2113, 2377, 2389, 2423, 2437, 2477

Offset: 1

William Lewis Craig, Mar 01 2017

nonn

For a given prime p, it has been proved that the set of all n for which p^2 divides n^n + (-1)^n (n-1)^(n-1) is some set of residue classes mod p(p-1). Therefore testing all values of n up to p(p-1) will determine whether p is in this list.

There are far more efficient ways to determine if p is indeed in the list, described by Boyd, Martin, and Thom in their paper.

William Lewis Craig, Table of n, a(n) for n = 1..10052
David W. Boyd, Greg Martin, and Mark Thom, Squarefree values of trinomial discriminants, arXiv 1402.5148 [math.NT], 2014.

Values of n for which square divisors occur are A238194.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[AnyTrue[Range[2, p(p-1)], Mod[PowerMod[#, #, p^2] + (-1)^# PowerMod[#-1, #-1, p^2], p^2] == 0&], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

isok(p) = {for (n=2, p*(p-1), if (((n^n + (-1)^n*(n-1)^(n-1)) % p^2) == 0, return (1)););}
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))) \\ Michel Marcus, Aug 01 2017

cp's OEIS Frontend

A283146 Prime numbers p whose square divides a number of the form n^n + (-1)^n (n-1)^(n-1), where n is a positive integer.

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