

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

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.


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



STATUS

approved



