A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).
7, 6, 8, 7, 1, 8, 3, 6, 2, 4, 4, 6, 4, 8, 5, 1, 9, 8, 6, 7, 2, 7, 3, 4, 3, 3, 2, 4, 5, 5, 3, 5, 0, 5, 2, 5, 2, 3, 4, 2, 5, 5, 7, 4, 0, 4, 1, 1, 9, 0, 4, 1, 1, 0, 7, 0, 1, 5, 4, 1, 3, 5, 2, 9, 3, 4, 8, 6, 0, 7, 7, 6, 8, 3, 3, 7, 9, 0, 8, 0, 3, 9, 3, 3, 2, 8, 8, 0, 7, 6, 4, 8, 9, 6, 9, 1, 4, 7, 5, 9, 5, 3, 3, 7, 2, 4
Offset: 0
Examples
0.76871836244648519867273433245535052523425574041190...
Links
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y3).
- V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 3.8-3.9, p. 1352-1353).
- László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.9, p. 29).
Programs
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Mathematica
$MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)/p*Sum[1/p^k/(p^k + 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]];RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
Formula
From Amiram Eldar, Dec 23 2024: (Start)
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k+1))).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A063974(k). (End)
Extensions
More digits from Vaclav Kotesovec, Jun 13 2021