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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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A141232, A137576, A001262, A020229, A062117, A006694.

Sequence in context: A048950 A329705 A020229 * A235408 A190877 A293566

Adjacent sequences:  A141347 A141348 A141349 * A141351 A141352 A141353

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

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Last modified December 1 06:09 EST 2020. Contains 338833 sequences. (Running on oeis4.)