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%I #30 Apr 16 2023 20:50:49
%S 3,1093,11,1093,20771,66161,5,3,11,3,71,2693,863,29,29131,1093,3,5,3,
%T 281
%N Smallest odd prime p such that p^2 | n^(p-1) - 1.
%C Similar to the sequence A039951 where p=2 is allowed.
%C a(n^k) <= a(n) for any n,k>1.
%C a(21) > 1.63*10^14 (see Fischer's link).
%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 {1, 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/FermatQuotient.html">Fermat quotient</a>
%H Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/">Fermat quotients B^(P-1) == 1 (mod P^2)</a>
%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.
%F a(n) = A039951(n) for all n not of the form 4k+1, while a(4k+1) > A039951(4k+1) = 2. - _Alexander Adamchuk_, Dec 03 2006
%t f[n_] := Block[{k = 2}, While[k < 5181800 && PowerMod[n, Prime[k] - 1, Prime[k]^2] != 1, k++ ]; If[k == 5181800, 0, Prime[k]]]; Table[ f[n], {n, 70}] (* _Robert G. Wilson v_, Jul 23 2004 *)
%o (PARI) for(n=2, 20, forprime(p=3, 1e9, if(Mod(n, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ _Felix Fröhlich_, Jul 24 2014
%Y Cf. A007663, A001220, A039951, A124121, 124122.
%K nonn,more,hard
%O 1,1
%A _Lekraj Beedassy_, Jul 22 2004
%E Definition corrected by _Alexander Adamchuk_, Nov 27 2006
%E Edited by _Max Alekseyev_, Oct 07 2009
%E Edited and updated by _Max Alekseyev_, Jan 29 2012
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