

A328663


Super pseudoprimes to base 3 (A328662) with more than two prime factors (counted with multiplicity).


4



7381, 512461, 532171, 1018601, 2044657, 3882139, 5934391, 8624851, 10802017, 14396449, 19383673, 25708453, 32285041, 35728129, 35807461, 38316961, 43040161, 53369149, 58546753, 59162891, 64464919, 71386849, 75397891, 79511671, 81276859, 83083001, 84890737, 85636609
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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, SpringerVerlag, New York, 2001, chapter 12, Fermat's Little Theorem, Pseudoprimes, and Superpseudoprimes, pp. 130146.


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. 157159, 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. 125129, entire volume.
Andrzej Rotkiewicz, Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci numbers, Springer, Dordrecht, 1999, pp. 293306.
Lawrence Somer, On superpseudoprimes, Mathematica Slovaca, Vol. 54, No. 5 (2004), pp. 443451.


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

Cf. A178997
Subsequence of A005935, A328662.
Sequence in context: A043806 A043824 A277286 * A022255 A189504 A028540
Adjacent sequences: A328660 A328661 A328662 * A328664 A328665 A328666


KEYWORD

nonn


AUTHOR

Amiram Eldar, Oct 24 2019


STATUS

approved



