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A085014
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For p = prime(n), a(n) is the number of primes q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq.
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3
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0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 1, 3, 2, 2, 4, 1, 2, 3, 5, 4, 3, 6, 4, 4, 6, 4, 5, 4, 6, 5, 4, 2, 5, 8, 7, 5, 6, 3, 3, 3, 4, 5, 4, 4, 5, 9, 8, 7, 8, 5, 8, 7, 8, 4, 6, 6, 7, 7, 9, 6, 11, 7, 8, 2, 7, 12, 8, 6, 8, 4, 5, 5, 6, 5, 11, 10, 9, 11, 5, 8, 9, 12, 9, 4, 7, 13, 8, 5
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,8
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COMMENTS
| Using a construction in Erdos' paper, it can be shown that a(prime(n)) > 0, except for the primes 3, 5, 7 and 13. Using a theorem of Lehmer, it can be shown that the possible values of q are among the prime factors of 2^(p-1)-1. The sequence A085012 gives the smallest prime q such that q*prime(n) is a pseudoprime.
Sequence A086019 gives the largest prime q such that q*prime(n) is a pseudoprime.
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REFERENCES
| P. Erdos, On the converse of Fermat's theorem, Amer. Math. Monthly 56 (1949), p. 623-624.
D. H. Lehmer, On the converse of Fermat's theorem, Amer. Math. Monthly 43 (1936), p. 347-354.
P. Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 105-112.
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EXAMPLE
| a(11) = 3 because prime(11) = 31 and 2^30-1 has 3 factors (11, 151, 331) that yield pseudoprimes when multiplied by 31.
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MATHEMATICA
| Table[p=Prime[n]; q=Transpose[FactorInteger[2^(p-1)-1]][[1]]; cnt=0; Do[If[PowerMod[2, p*q[[i]]-1, p*q[[i]]]==1, cnt++ ], {i, Length[q]}]; cnt, {n, 2, 50}]
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CROSSREFS
| Cf. A001567 (base-2 pseudoprimes), A085012, A086019, A180471.
Sequence in context: A131756 A194859 A194838 * A082074 A132283 A088370
Adjacent sequences: A085011 A085012 A085013 * A085015 A085016 A085017
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jun 28 2003
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