A141216 a(n) = A137576((N-1)/2) - N, where N = A001567(n).

30, 320, 224, 240, 72, 360, 728, 0, 672, 216, 1320, 0, 0, 16, 5060, 60, 126, 10560, 216, 0, 3360, 2574, 150, 5040, 2808, 3600, 3600, 232, 400, 420, 22, 2700, 2784, 224, 96, 70, 1640, 240, 9200, 3600, 2760, 58344, 616, 504, 102, 5600, 8064, 264, 11880, 1440, 7488, 252

Offset: 1

Author

Vladimir Shevelev, Jun 14 2008, Jul 13 2008

Keywords

• nonn

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.

Crossrefs

Cf. A137576, A001567, A001262, A002326, A006694.

Programs

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 *)

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

Extensions

More terms via b137576.txt from R. J. Mathar, Jun 12 2010

More terms from Michel Marcus, Dec 09 2018