A367325 Decimal expansion of Sum_{k>=1} sigma(k)/(2^k-1), where sigma(k) is the sum of divisors of k (A000203).
3, 6, 0, 4, 4, 4, 7, 3, 4, 1, 9, 7, 1, 9, 4, 6, 7, 4, 4, 8, 9, 3, 6, 4, 8, 4, 7, 3, 6, 2, 3, 5, 8, 8, 3, 5, 6, 0, 0, 4, 9, 5, 4, 8, 7, 0, 6, 4, 9, 9, 8, 7, 5, 0, 1, 3, 8, 3, 6, 2, 6, 3, 1, 5, 0, 9, 6, 6, 2, 9, 5, 0, 1, 7, 0, 7, 1, 3, 4, 9, 6, 6, 9, 1, 7, 8, 2, 2, 8, 9, 9, 1, 6, 9, 9, 2, 7, 5, 4, 4, 2, 7, 0, 3, 8
Offset: 1
Examples
3.60444734197194674489364847362358835600495487064998...
Links
- Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020. See eq. (6.1c), p. 13.
- Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 526.
Programs
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Maple
with(numtheory): evalf(sum(sigma(k)/(2^k-1), k = 1..infinity), 120)
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Mathematica
RealDigits[Sum[DivisorSigma[1,n]/(2^n-1), {n, 1, 500}], 10, 120][[1]] -
PARI
suminf(k = 1, sigma(k)/(2^k-1))
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PARI
suminf(k = 1, numdiv(k)/((2^k-1)*(1-1/2^k)))
Formula
Equals Sum_{k>=1} d(k)/((2^k-1)*(1-1/2^k)), where d(k) is the number of divisors of k (A000005).