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A039951 a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1. 40

%I #87 Apr 03 2023 10:36:09

%S 2,1093,11,1093,2,66161,5,3,2,3,71,2693,2,29,29131,1093,2,5,3,281,2,

%T 13,13,5,2,3,11,3,2,7,7,5,2,46145917691,3,66161,2,17,8039,11,2,23,5,3,

%U 2,3

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

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

%C 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

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

%C 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

%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=FermatQuotient">Fermat quotient</a>.

%H Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/">Fermat quotients B^(P-1) == 1 (mod P^2)</a>

%H Richard Fischer, <a href="/A039951/a039951_1.txt">Update Table of n</a>, July 15 2021.

%H W. Keller and J. Richstein, <a href="http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients q_p(a) that are divisible by p</a>.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_762.htm">Puzzle 762. Conjecture from Ribenboim's book</a>, The Prime Puzzles and Problems Connection.

%H Robert G. Wilson v, <a href="/A039951/a039951.txt">Table of n, a(n) for n = 1..10000 (with missing terms)</a>

%F a(4k+1) = 2.

%F 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]

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

%t 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 *)

%o (PARI) for(n=1, 20, forprime(p=2, 1e9, if(Mod(n, p^2)^(p-1)==1), print1(p, ", "); next({2}))); print1("--, ")) \\ _Felix Fröhlich_, Jul 24 2014

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

%K nonn,more,hard

%O 1,1

%A _David W. Wilson_

%E a(34)-a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004

%E Entry revised by _N. J. A. Sloane_, Nov 30 2006

%E Edited by _Max Alekseyev_, Oct 06, Oct 09 2009

%E Edited and updated by _Max Alekseyev_, Jan 29 2012

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)