A328664 Least super pseudoprime to base n that is not a semiprime.

294409, 7381, 13981, 342271, 9331, 747289, 63, 8, 99, 4921, 1729, 12, 195, 355957, 255, 8, 325, 18, 399, 20, 483, 1183, 575, 8, 27, 1729, 27, 28, 637, 30, 1023, 8, 105, 153, 1295, 12, 1105, 29659, 1599, 8, 12167, 42, 45, 44, 45, 1105, 637, 8, 147, 50, 2703, 27

Offset: 2

Author

Amiram Eldar, Oct 24 2019

Keywords

• nonn

Comments

A number is super pseudoprime to base n > 1 if it is a Fermat pseudoprime to base n and of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base n.

The semiprime Fermat pseudoprimes are trivial terms since they do not have composite proper divisors.

Examples

			a(2) = 294409 = 37 * 73 * 109 is the first term of A178997.
a(3) = 7381 = 11^2 * 61 is the first term of A328663.
		

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 = 2..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.

Crossrefs

Cf. A178997, A328662, A328663.

Programs

Mathematica

a[n_] := Module[{k=1}, While[PrimeOmega[k] < 3 || !AllTrue[Rest[Divisors[k]], PowerMod[n, #-1, #] == 1 &], k++]; k]; Array[a, 10, 2]