A141390 Overpseudoprimes to base 5.

781, 1541, 5461, 13021, 15751, 25351, 29539, 38081, 40501, 79381, 100651, 121463, 133141, 195313, 216457, 315121, 318551, 319507, 326929, 341531, 353827, 375601, 416641, 432821, 453331, 464881, 498451, 555397, 556421, 753667, 764941, 863329, 872101, 886411

Offset: 1

Author

Vladimir Shevelev, Jun 29 2008

Keywords

• nonn

Comments

If h_5(n) is the multiplicative order of 5 modulo n, r_5(n) is the number of cyclotomic cosets of 5 modulo n then, by the definition, n is an overpseudoprime of base 5 if h_5(n)*r_5(n)+1=n. These numbers are in A020231. In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime to base 5 iff h_5(p_1)=...=h_5(p_k). E.g., since h_5(101)=h_5(251)=h_5(401)=25, the number 101*251*401=10165751 is in the sequence.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..327 (calculated from the b-file at A020231)
• V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arXiv:0806.3412 [math.NT], 2008-2012.
• V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv preprint arXiv:1206:0606, 2012. - From _N. J. A. Sloane_, Oct 28 2012
• V. 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.

Crossrefs

Cf. A141232, A141350, A020231, A020229.

Programs

Mathematica

ops5Q[n_] := CompositeQ[n] && GCD[n, 5] == 1 && MultiplicativeOrder[5, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[5, #] &] - 1) + 1 == n; Select[Range[6, 10^6], ops5Q] (* Amiram Eldar, Jun 24 2019 *)

PARI

isok(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018

Extensions

Inserted a(2) and a(8) and extended at the suggestion of Gilberto Garcia-Pulgarin by Vladimir Shevelev, Feb 06 2012