A039951 - OEIS

A039951

a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1.

2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3

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OFFSET

1,1

COMMENTS

a(n^k) <= a(n) for any n,k > 1.

a(n) is currently unknown for n in {47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, ...}. - Richard Fischer, Jul 15 2021

a(47) > 1.4*10^14, a(72) > 1.4*10^14 (see Fischer's tables).

a(n) > 10^14 for n = 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, extending the search bound for these bases from 4.4*10^12. - Daniel Okwor, Jun 07 2026

For all nonnegative integers n and k, a(n^(n^k)) = a(n) (see Puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014

LINKS

Table of n, a(n) for n=1..46.

C. K. Caldwell, The Prime Glossary, Fermat quotient.

Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)

Richard Fischer, Update Table of n, July 15 2021.

W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.

Daniel Okwor, Verification that no Wieferich prime exists below 10^14 for 27 bases, 2026.

Carlos Rivera, Puzzle 762. Conjecture from Ribenboim's book, The Prime Puzzles and Problems Connection.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (with missing terms)

FORMULA

a(4k+1) = 2.

a(n) = A096082(n) for all n > 1 that are not of the form 4k+1. Note that A096082 begins with n = 2. [Corrected and clarified by Jonathan Sondow, Jun 17-18 2010]

MATHEMATICA

Table[p = 2; While[! Divisible[n^(p - 1) - 1, p^2], p = NextPrime@ p]; p, {n, 33}] (* Michael De Vlieger, Nov 24 2016 *)

f[n_] := Block[{p = 2}, While[ PowerMod[n, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 33] (* Robert G. Wilson v, Jul 18 2018 *)

PROG

(PARI) a(n)={forprime(p=2, oo, if(Mod(n, p^2)^(p-1)==1, return(p))); oo} \\ Felix Fröhlich, Jul 24 2014

CROSSREFS

Cf. A001220, A045616, A096082, A014127, A123692, A123693, A174422.

Sequence in context: A321633 A244550 A359952 * A247072 A282293 A252358

Adjacent sequences: A039948 A039949 A039950 * A039952 A039953 A039954

KEYWORD

nonn,more,hard,changed

AUTHOR