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A086019
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For p = prime(n), a(n) is the largest prime q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq; a(n) is 0 if no such prime exists.
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2
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0, 0, 0, 31, 0, 257, 73, 683, 113, 331, 109, 61681, 5419, 2796203, 1613, 3033169, 1321, 599479, 122921, 38737, 22366891, 8831418697, 2931542417, 22253377, 268501, 131071, 28059810762433, 279073, 54410972897, 77158673929, 145295143558111
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,4
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COMMENTS
| 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. Sequence A085012 gives the smallest prime q such that 2^(pq-1) = 1 mod pq. Sequence A085014 gives the number of 2-factor pseudoprimes that have prime(n) as a factor.
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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(9) = 683 because prime(9) = 23 and 683 is the largest factor of 2^22-1 that yields a pseudoprime when multiplied by 23.
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MATHEMATICA
| Table[p=Prime[n]; q=Reverse[Transpose[FactorInteger[2^(p-1)-1]][[1]]]; i=1; While[i<=Length[q]&&(PowerMod[2, p*q[[i]]-1, p*q[[i]]]>1), i++ ]; If[i>Length[q], 0, q[[i]]], {n, 2, 40}]
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CROSSREFS
| Cf. A001567 (base 2 pseudoprimes), A085012, A085014, A180471.
Sequence in context: A140762 A028363 A085012 * A040968 A040967 A040965
Adjacent sequences: A086016 A086017 A086018 * A086020 A086021 A086022
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KEYWORD
| hard,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jul 08 2003
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