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A085014 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. 5
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; text; internal format)
OFFSET

2,8

COMMENTS

Using a construction in Erdős's 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.

REFERENCES

Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 105-112.

LINKS

Amiram Eldar, Table of n, a(n) for n = 2..337

Paul Erdős, 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.

Index entries for sequences related to pseudoprimes

FORMULA

a(n) < 0.7 * p, where p is the n-th prime. - Charles R Greathouse IV, Apr 12 2012

EXAMPLE

a(11) = 3 because prime(11) = 31 and 2^30-1 has 3 prime factors (11, 151, 331) that yield pseudoprimes when multiplied by 31.

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}]

CROSSREFS

Cf. A001567 (base-2 pseudoprimes), A085012, A086019, A180471.

Sequence in context: A212620 A194859 A194838 * A082074 A132283 A307081

Adjacent sequences:  A085011 A085012 A085013 * A085015 A085016 A085017

KEYWORD

nonn

AUTHOR

T. D. Noe, Jun 28 2003

STATUS

approved

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Last modified July 31 13:25 EDT 2021. Contains 346373 sequences. (Running on oeis4.)