OFFSET
1,1
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jun 29 2008
EXTENSIONS
Inserted a(2) and a(8) and extended at the suggestion of Gilberto Garcia-Pulgarin by Vladimir Shevelev, Feb 06 2012
STATUS
approved