2,1

Smallest prime p such that p^2 divides n^(p-1) - 1, or 0 if no such p exists, are listed in A039951(n) = {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,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 03 2006

a(n) = A039951(n) for all n that are not of the form 4k+1. A039951(4k+1) = 2. - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 03 2006

Note that a(n) = 1093 for n = {2,4,16,256}, a(n) = 66161 for n = {6,36,216}, a(n) = 2693 for n = {12,144}, a(n) = 29 for n = {14,196}, a(n) = 281 for n = {20,400}. It appears that there is a pattern such that in many cases if a(n) = p then a(n^2) = p too. In some cases as for n = 6 a(n^3) = p too. - Alexander Adamchuk (alex(AT)kolmogorov.com), Dec 03 2006

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

W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p

W. Keller and J. Richstein, Solutions of a^{p-1} == 1 (mod p^2) for all a <= 1000 and odd primes p < 10^10

W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.

Helmut Richter, Table of all known a(n) up to n = 1000

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}] (from Robert G. Wilson v Jul 23 2004)

Cf. A007663, A001220.

Cf. A039951 = Smallest prime p such that p^2 divides n^(p-1) - 1, or 0 if no such p exists. Cf. A124121, 124122.

Sequence in context: A043864 A043873 A091673 this_sequence A138698 A023698 A038469

Adjacent sequences: A096079 A096080 A096081 this_sequence A096083 A096084 A096085

nonn,more

Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 22 2004

Definition corrected by Alexander Adamchuk, Nov 27 2006