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A328662
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Super pseudoprimes (or superpseudoprimes) to base 3: Fermat pseudoprimes to base 3 all of whose divisors that are larger than 1 are either primes or Fermat pseudoprimes to base 3.
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5
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91, 121, 671, 703, 949, 1541, 1891, 2701, 3281, 7381, 8401, 12403, 14383, 15203, 16531, 18721, 23521, 24727, 28009, 30857, 31621, 31697, 38503, 44287, 46999, 47197, 49051, 49141, 55261, 55969, 63139, 72041, 74593, 79003, 82513, 83333, 88573, 88831, 90751, 96139
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OFFSET
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1,1
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COMMENTS
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The super pseudoprimes to base 2 are the super-Poulet numbers (A050217).
Includes all the semiprimes in A005935. The first terms that are not semiprimes are 7381, 512461, 532171, 1018601, ... (A328663).
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REFERENCES
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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.
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LINKS
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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.
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EXAMPLE
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91 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 7 and 13.
7381 is in the sequence since it is a Fermat pseudoprime to base 3, and its proper divisors that are larger than 1 are the primes 11 and 61, and the composite numbers 121 and 671 that are Fermat pseudoprimes to base 3.
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MATHEMATICA
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aQ[n_]:= CompositeQ[n] && AllTrue[Rest[Divisors[n]], PowerMod[3, #-1, #] == 1 &]; Select[Range[10^5], aQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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