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A141390 Overpseudoprimes to base 5. 3
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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A115467 A338877 A020231 * A038477 A236888 A006113
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

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)