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

1,1

Comments

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.

Links

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.

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Values of n for which square divisors occur are A238194.

Sequence in context: A026050 A347804 A304356 * A068209 A139958 A097459

Adjacent sequences: A283143 A283144 A283145 * A283147 A283148 A283149

Keyword

nonn

Author

William Lewis Craig, Mar 01 2017

Status

approved