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.
91, 121, 671, 703, 949, 1541, 1891, 2701, 3281, 7381, 8401, 12403, 14383, 15203, 16531, 18721, 23521, 24727, 28009, 30857, 31621, 31697, 38503, 44287, 46999, 47197, 49051, 49141, 55261, 55969, 63139, 72041, 74593, 79003, 82513, 83333, 88573, 88831, 90751, 96139
Offset: 1
Keywords
Examples
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.
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.
Programs
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Mathematica
aQ[n_]:= CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
Comments