A141232 Overpseudoprimes to base 2: composite k such that k = A137576((k-1)/2).

2047, 3277, 4033, 8321, 65281, 80581, 85489, 88357, 104653, 130561, 220729, 253241, 256999, 280601, 390937, 458989, 486737, 514447, 580337, 818201, 838861, 877099, 916327, 976873, 1016801, 1082401, 1145257, 1194649, 1207361, 1251949, 1252697, 1325843

Offset: 1

Author

Vladimir Shevelev, Jun 16 2008

Keywords

• nonn

Comments

Numbers are found by prime factorization of numbers from A001262 and a simple testing of the conditions indicated in comment to A141216.

All composite Mersenne numbers (A001348), Fermat numbers (A000215) and squares of Wieferich primes (A001220) are in this sequence. - Vladimir Shevelev, Jul 15 2008

C. Pomerance proved that this sequence is infinite (see Theorem 4 in the third reference). - Vladimir Shevelev, Oct 29 2011

Odd composite numbers k such that ord(2,k) * ((Sum_{d|k} phi(d)/ord(2,d)) - 1) = k-1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..664 from Michel Marcus)
• J. H. Castillo, G. García-Pulgarín and J. M. Velásquez-Soto, q-pseudoprimality: A natural generalization of strong pseudoprimality, arXiv:1412.5226 [math.NT], 2014.
• Vladimir Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich primes, arXiv:0806.3412 [math.NT], 2008-2012.
• Vladimir Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.
• Vladimir Shevelev, On upper estimate for the overpseudoprime counting function, arXiv:0807.1975 [math.NT], 2008.
• Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.
• Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv:1206.0606 [math.NT], 2012.

Crossrefs

Cf. A000010, A001262, A141216, A001567, A002326.

Programs

Mathematica

A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #]&] - t + 1];
okQ[n_] := n > 1 && CompositeQ[n] && n == A137576[(n-1)/2];
Reap[For[k = 2, k < 2*10^6, k++, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jan 11 2019, from PARI *)

PARI

f(n)=my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(n) = (n>1) && !isprime(n) && (n == f((n-1)/2)); \\ Michel Marcus, Oct 05 2018

Formula

Sum_{n:a(n)<=x} 1 <= x^(3/4)(1+o(1)).

Extensions

Name edited by Michel Marcus, Oct 05 2018