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A092816
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Number of Sophie Germain primes less than 10^n.
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9
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3, 10, 37, 190, 1171, 7746, 56032, 423140, 3308859, 26569515, 218116524, 1822848478, 15462601989, 132822315652
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OFFSET
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1,1
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COMMENTS
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Hardy-Littlewood conjecture: Number of Sophie Germain primes less than n ~ 2*C2*n/(log(n))^2, where C2 = 0.6601618158... is the twin prime constant (see A005597). The truth of the above conjecture would imply that there are an infinite number of Sophie Germain primes (which is also conjectured). - Robert G. Wilson v, Jan 31 2013
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REFERENCES
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P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.
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LINKS
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FORMULA
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For 1 < n < 15, a(n) ~ e * (pi(2*10^n) - pi(10^n)) / (5*n - 5) where e is Napier's constant, see A001113 (we use n > 1 to avoid division by zero; whether the formula holds for any n > 14 is unknown). - Sergey Pavlov, Apr 07 2021 [This formula fails under the Hardy-Littlewood conjecture; the leading constant is wrong. - Charles R Greathouse IV, Aug 03 2023]
For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(2 * 10^n + 1) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that both c and c' = 2*c + 1 are composite (trivial). - Sergey Pavlov, Apr 08 2021
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EXAMPLE
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The Sophie Germain primes up to 10 are 2 (since 5 is prime), 3 (since 7 is prime), and 5 (since 11 is prime), so a(1) = 3.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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