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A328662 Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.

%I A328662 #16 Dec 04 2020 17:36:14
%S A328662 91,121,671,703,949,1541,1891,2701,3281,7381,8401,12403,14383,15203,
%T A328662 16531,18721,23521,24727,28009,30857,31621,31697,38503,44287,46999,
%U A328662 47197,49051,49141,55261,55969,63139,72041,74593,79003,82513,83333,88573,88831,90751,96139
%N A328662 Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.
%C A328662 The super pseudoprimes to base 2 are the super-Poulet numbers (A050217).
%C A328662 Includes all the semiprimes in A005935. The first terms that are not semiprimes are 7381, 512461, 532171, 1018601, ... (A328663).
%C A328662 Subsequence of A271116. - _Bill McEachen_, Nov 06 2020
%D A328662 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 A328662 Amiram Eldar, <a href="/A328662/b328662.txt">Table of n, a(n) for n = 1..10000</a>
%H A328662 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 A328662 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 A328662 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 A328662 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 A328662 91 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 7 and 13.
%e A328662 7381 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 11 and 61, and the composite numbers 121 and 671 that are Fermat pseudoprimes to base 3.
%t A328662 aQ[n_]:=  CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
%Y A328662 Subsequence of A005935.
%Y A328662 Cf. A050217.
%K A328662 nonn
%O A328662 1,1
%A A328662 _Amiram Eldar_, Oct 24 2019