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%I #9 Oct 25 2019 05:01:16
%S 7381,512461,532171,1018601,2044657,3882139,5934391,8624851,10802017,
%T 14396449,19383673,25708453,32285041,35728129,35807461,38316961,
%U 43040161,53369149,58546753,59162891,64464919,71386849,75397891,79511671,81276859,83083001,84890737,85636609
%N Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).
%C Super pseudoprimes to base 3 are Fermat pseudoprimes to base 3 all of whose composite divisors are also Fermat pseudoprimes to base 3. Therefore all the Fermat pseudoprimes to base 3 that are semiprimes are super pseudoprimes. This sequence contains the nontrivial terms of A328662, i.e. terms with at least one composite proper divisor.
%C Fehér and Kiss proved that there are infinitely many terms with 3 distinct prime factors (their proof was for all bases a > 1 that are not divisible by 4. Phong proved it for all bases a > 1).
%C The first term, 7381, is not squarefree. What is the next such term?
%D Michal Krížek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer-Verlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130-146.
%H Amiram Eldar, <a href="/A328663/b328663.txt">Table of n, a(n) for n = 1..400</a>
%H J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, <a href="http://annalesm.elte.hu/annales26-1983/Annales_1983_T-XXVI.pdf">entire volume</a>.
%H B. M. Phong, On super pseudoprimes which are products of three primes, Ann. Univ. Sci. Budapest. Eótvós Sect. Math., Vol. 30 (1987), pp. 125-129, <a href="http://annalesm.elte.hu/annales30-1987/Annales_1987_T-XXX.pdf">entire volume</a>.
%H Andrzej Rotkiewicz, <a href="https://doi.org/10.1007/978-94-011-4271-7_28">Solved and unsolved problems on pseudoprime numbers and their generalizations</a>, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
%H Lawrence Somer, <a href="https://dml.cz/handle/10338.dmlcz/130094">On superpseudoprimes</a>, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.
%e 512461 is in the sequence since it is a Fermat pseudoprime to base 3, 3^512460 == 1 (mod 512461), and all of its divisors that are larger than 1 are either primes (31, 61, and 271), or Fermat pseudoprimes to base 3 (1891, 8401, 16531, 512461).
%t aQ[n_]:= PrimeOmega[n] > 2 && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
%Y Cf. A178997
%Y Subsequence of A005935, A328662.
%K nonn
%O 1,1
%A _Amiram Eldar_, Oct 24 2019
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