OFFSET
1,1
COMMENTS
The super pseudoprimes to base 2 are the super-Poulet numbers (A050217).
Includes all the semiprimes in A005935. The first terms that are not semiprimes are 7381, 512461, 532171, 1018601, ... (A328663).
Subsequence of A271116. - Bill McEachen, Nov 06 2020
REFERENCES
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Fehér and P. Kiss, Note on super pseudoprime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983), pp. 157-159, entire volume.
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, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293-306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443-451.
EXAMPLE
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.
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.
MATHEMATICA
aQ[n_]:= CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 24 2019
STATUS
approved