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A328013
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Decimal expansion of the growth constant for the partial sums of powerful part of n (A057521).
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2
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3, 5, 1, 9, 5, 5, 5, 0, 5, 8, 4, 1, 7, 1, 0, 6, 6, 4, 7, 1, 9, 7, 5, 2, 9, 4, 0, 3, 6, 9, 8, 5, 7, 8, 1, 7, 1, 8, 6, 0, 3, 9, 8, 0, 8, 2, 2, 5, 4, 0, 7, 8, 1, 4, 7, 1, 1, 4, 6, 4, 0, 3, 1, 4, 5, 4, 1, 7, 8, 3, 9, 8, 4, 7, 9, 7, 3, 5, 4, 0, 8, 9, 7, 7, 1, 3, 5, 8, 0, 3, 7, 5, 3, 6, 4, 6, 1, 6, 2, 0, 1, 1, 4, 5, 5
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OFFSET
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1,1
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LINKS
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FORMULA
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The constant d1 in the paper by Cloutier et al. such that Sum_{k=1..x} A057521(k) = (d1/3)*x^(3/2) + O(x^(4/3)).
Sum_{k=1..x} 1/A055231(k) = d1*x^(1/2) + O(x^(1/3)).
Equals Product_{primes p} (1 + 2/p^(3/2) - 1/p^(5/2)).
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EXAMPLE
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3.51955505841710664719752940369857817186039808225407...
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MATHEMATICA
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$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 0, -2, 0, 1}, {0, 0, 6, 0, -5}, m]; RealDigits[(1 + 2/2^(3/2) - 1/2^(5/2))*(1 + 2/3^(3/2) - 1/3^(5/2))* Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
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PROG
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(PARI) prodeulerrat(1 + 2/p^3 - 1/p^5, 1/2) \\ Amiram Eldar, Jun 29 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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