OFFSET
1,1
COMMENTS
Super pseudoprimes to base 3 are Fermat pseudoprimes to base 3 all of whose composite divisors are also Fermat pseudoprimes to base 3. Therefore all the Fermat pseudoprimes to base 3 that are semiprimes are super pseudoprimes. This sequence contains the nontrivial terms of A328662, i.e. terms with at least one composite proper divisor.
Fehér and Kiss proved that there are infinitely many terms with 3 distinct prime factors (their proof was for all bases a > 1 that are not divisible by 4. Phong proved it for all bases a > 1).
The first term, 7381, is not squarefree. What is the next such term?
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..400
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
512461 is in the sequence since it is a Fermat pseudoprime to base 3, 3^512460 == 1 (mod 512461), and all of its divisors that are larger than 1 are either primes (31, 61, and 271), or Fermat pseudoprimes to base 3 (1891, 8401, 16531, 512461).
MATHEMATICA
aQ[n_]:= PrimeOmega[n] > 2 && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 24 2019
STATUS
approved