

A039951


a(n) is the smallest prime p such that p^2 divides n^(p1)  1.


40



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).
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^(P1) == 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.
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 1718 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) for(n=1, 20, forprime(p=2, 1e9, if(Mod(n, p^2)^(p1)==1), print1(p, ", "); next({2}))); print1(", ")) \\ Felix FrÃ¶hlich, Jul 24 2014


CROSSREFS

Cf. A001220, A045616, A096082, A014127, A123692, A123693, A174422.
Sequence in context: A344669 A321633 A244550 * A247072 A282293 A252358
Adjacent sequences: A039948 A039949 A039950 * A039952 A039953 A039954


KEYWORD

nonn,more,hard


AUTHOR

David W. Wilson


EXTENSIONS

a(34)a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004
Entry revised by N. J. A. Sloane, Nov 30 2006
Edited by Max Alekseyev, Oct 06, Oct 09 2009
Edited and updated by Max Alekseyev, Jan 29 2012


STATUS

approved



