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%I #23 Jun 28 2019 14:40:33
%S 30,320,224,240,72,360,728,0,672,216,1320,0,0,16,5060,60,126,10560,
%T 216,0,3360,2574,150,5040,2808,3600,3600,232,400,420,22,2700,2784,224,
%U 96,70,1640,240,9200,3600,2760,58344,616,504,102,5600,8064,264,11880,1440,7488,252
%N a(n) = A137576((N-1)/2) - N, where N = A001567(n).
%C 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.
%C 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.
%C Note that in A001262 there exist terms which are not squarefree. The first is A001262(52) = 1194649 = 1093^2.
%C 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.
%H Amiram Eldar, <a href="/A141216/b141216.txt">Table of n, a(n) for n = 1..10000</a>
%H V. Shevelev, <a href="http://arxiv.org/abs/0806.3412">Overpseudoprimes, Mersenne Numbers and Wieferich Primes</a>, arxiv:0806.3412 [math.NT], 2008-2012.
%t 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 *)
%o (PARI) f(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
%o isfpp(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1}; \\ A001567
%o lista(nn) = {for (n=1, nn, if (isfpp(n), print1(f((n-1)/2) - n, ", ");););} \\ _Michel Marcus_, Dec 09 2018
%Y Cf. A137576, A001567, A001262, A002326, A006694.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Jun 14 2008, Jul 13 2008
%E More terms via b137576.txt from _R. J. Mathar_, Jun 12 2010
%E More terms from _Michel Marcus_, Dec 09 2018
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