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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A322919 A026050 A304356 * A068209 A139958 A097459

Adjacent sequences:  A283143 A283144 A283145 * A283147 A283148 A283149

KEYWORD

nonn

AUTHOR

William Lewis Craig, Mar 01 2017

STATUS

approved

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Last modified March 28 12:31 EDT 2020. Contains 333085 sequences. (Running on oeis4.)