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
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
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