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A141216 a(n) = A137576((N-1)/2) - N, where N = A001567(n). 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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

Sequence in context: A042750 A074994 A134287 * A159543 A227689 A006859

Adjacent sequences:  A141213 A141214 A141215 * A141217 A141218 A141219

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

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Last modified August 22 00:43 EDT 2019. Contains 326169 sequences. (Running on oeis4.)