OFFSET
1,1
COMMENTS
The zero terms are of a special interest. Indeed, since for any odd prime p, A137576((p-1)/2)=p, then it is natural to call "overpseudoprimes" those Poulet pseudoprimes A001567(n) for which a(n)=0.
Theorem. A squarefree composite number m = p_1*p_2*...*p_k is an overpseudoprime if and only if A002326((p_1-1)/2) = A002326((p_2-1)/2) = ... = A002326((p_k-1)/2). Moreover, every overpseudoprime is in A001262.
Note that in A001262 there exist terms which are not squarefree. The first is A001262(52) = 1194649 = 1093^2.
It can be shown that if an overpseudoprime is not a multiple of the square of a Wieferich prime (see A001220) then it is squarefree. Also all squares of Wieferich primes are overpseudoprimes.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv:0806.3412 [math.NT], 2008-2012.
MATHEMATICA
fppQ[n_]:=PowerMod[2, n, n]==2; f[n_] := (t = MultiplicativeOrder[2, 2n+1])*DivisorSum[2n+1, EulerPhi[#] / MultiplicativeOrder[2, #]&]-t+1; s={}; Do[If[fppQ[n] && CompositeQ[n], AppendTo[s, f[(n-1)/2 ]-n]], {n, 1, 10000}]; s (* Amiram Eldar, Dec 09 2018 after Jean-François Alcover at A137576 *)
PROG
(PARI) f(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isfpp(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1}; \\ A001567
lista(nn) = {for (n=1, nn, if (isfpp(n), print1(f((n-1)/2) - n, ", "); ); ); } \\ Michel Marcus, Dec 09 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jun 14 2008, Jul 13 2008
EXTENSIONS
More terms via b137576.txt from R. J. Mathar, Jun 12 2010
More terms from Michel Marcus, Dec 09 2018
STATUS
approved