A096082 Smallest odd prime p such that p^2 | n^(p-1) - 1.

3, 1093, 11, 1093, 20771, 66161, 5, 3, 11, 3, 71, 2693, 863, 29, 29131, 1093, 3, 5, 3, 281

Offset

1,1

Comments

Similar to the sequence A039951 where p=2 is allowed.

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

a(21) > 1.63*10^14 (see Fischer's link).

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

Links

Table of n, a(n) for n=1..20.

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

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

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.

Formula

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A007663, A001220, A039951, A124121, 124122.

Sequence in context: A004803 A065604 A062612 * A274994 A178195 A199236

Adjacent sequences: A096079 A096080 A096081 * A096083 A096084 A096085

Keyword

nonn,more,hard

Author

Lekraj Beedassy, Jul 22 2004

Extensions

Definition corrected by Alexander Adamchuk, Nov 27 2006

Edited by Max Alekseyev, Oct 07 2009

Edited and updated by Max Alekseyev, Jan 29 2012

Status

approved