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A141350
Overpseudoprimes to base 3.
5
121, 703, 3281, 8401, 12403, 31621, 44287, 47197, 55969, 74593, 79003, 88573, 97567, 105163, 112141, 211411, 221761, 226801, 228073, 293401, 313447, 320167, 328021, 340033, 359341, 432821, 443713, 453259, 478297, 497503, 504913, 679057, 709873, 801139, 867043, 894781, 973241, 1042417
OFFSET
1,1
COMMENTS
If h_3(n) is the multiplicative order of 3 modulo n, r_3(n) is the number of cyclotomic cosets of 3 modulo n then, by the definition, n is an overpseudoprime to base 3 if h_3(n)*r_3(n)+1=n. These numbers are in A020229.
In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime of base 3 iff h_3(p_1)=...=h_3(p_k).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10798 (calculated using the b-file at A020229)
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 [math.NT], 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
ops3Q[n_] := CompositeQ[n] && GCD[n, 3] == 1 && MultiplicativeOrder[3, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[3, #] &] - 1) + 1 == n; Select[Range[10^6], ops3Q] (* Amiram Eldar, Jun 24 2019 *)
PROG
(PARI) isok(n) = (n!=1) && !isprime(n) && (gcd(n, 3)==1) && (znorder(Mod(3, n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(3, d))) - 1) + 1 == n); \\ Michel Marcus, Oct 25 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jun 27 2008, corrected Sep 07 2008
EXTENSIONS
a(10)-a(38) from Gilberto Garcia-Pulgarin added by Vladimir Shevelev, Feb 06 2012
STATUS
approved