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

2,1

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.

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.

Example

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

Mathematica

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

Crossrefs

Cf. A178997, A328662, A328663.

Sequence in context: A050249 A354440 A224973 * A328935 A335584 A182206

Adjacent sequences: A328661 A328662 A328663 * A328665 A328666 A328667

Keyword

nonn

Author

Amiram Eldar, Oct 24 2019

Status

approved