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A306071
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Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).
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11
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8, 0, 7, 3, 3, 0, 8, 2, 1, 6, 3, 6, 2, 0, 5, 0, 3, 9, 1, 4, 8, 6, 5, 4, 2, 7, 9, 9, 3, 0, 0, 3, 1, 1, 3, 4, 0, 2, 5, 8, 4, 5, 8, 2, 5, 0, 8, 1, 5, 5, 6, 6, 4, 4, 0, 1, 8, 0, 0, 5, 2, 0, 7, 7, 0, 4, 4, 1, 3, 8, 1, 4, 8, 4, 9, 3, 7, 5, 1, 8, 6, 4, 9, 6, 9, 5, 6, 0, 9, 3, 5, 0, 9, 6, 2, 9, 4, 8, 3, 7, 6, 5, 0, 1, 1, 8
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OFFSET
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0,1
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COMMENTS
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The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).
The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.
The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - Amiram Eldar, Aug 27 2019
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REFERENCES
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József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.
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LINKS
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László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 508), preprint, arXiv:1310.7053 [math.NT] (2014) (see p. 21).
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FORMULA
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Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).
Equals zeta(2) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4).
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EXAMPLE
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0.80733082163620503914...
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MATHEMATICA
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cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
$MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}], 120]]], {t, 300, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)
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PROG
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(PARI) prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021
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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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