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A141350 Overpseudoprimes to base 3. 5

%I

%S 121,703,3281,8401,12403,31621,44287,47197,55969,74593,79003,88573,

%T 97567,105163,112141,211411,221761,226801,228073,293401,313447,320167,

%U 328021,340033,359341,432821,443713,453259,478297,497503,504913,679057,709873,801139,867043,894781,973241,1042417

%N Overpseudoprimes to base 3.

%C 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.

%C 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).

%H Amiram Eldar, <a href="/A141350/b141350.txt">Table of n, a(n) for n = 1..10798</a> (calculated using the b-file at A020229)

%H V. Shevelev, <a href="http://arxiv.org/abs/0806.3412">Overpseudoprimes, Mersenne Numbers and Wieferich Primes</a>, arXiv:0806.3412 [math.NT], 2008-2012.

%H V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, <a href="http://arxiv.org/abs/1206.0606">Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers</a>, arXiv preprint arXiv:1206:0606 [math.NT], 2012. - From _N. J. A. Sloane_, Oct 28 2012

%H V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers</a>, J. Integer Seq. 15 (2012) Article 12.7.7.

%t 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 *)

%o (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

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

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Jun 27 2008, corrected Sep 07 2008

%E a(10)-a(38) from Gilberto Garcia-Pulgarin added by _Vladimir Shevelev_, Feb 06 2012

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)