A308039 Decimal expansion of lim_{i->oo} c(i)/i, where c(i) is the number of integers k such that sigma(k) < i (A074753).
6, 7, 2, 7, 3, 8, 3, 0, 9, 2, 1, 7, 4, 0, 9, 7, 9, 5, 3, 2, 7, 6, 8, 7, 2, 0, 3, 0, 8, 8, 9, 8, 6, 8, 6, 8, 9, 7, 0, 8, 7, 6, 8, 2, 9, 4, 1, 0, 2, 3, 2, 7, 3, 1, 2, 3, 5, 7, 1, 4, 5, 1, 8, 8, 2, 1, 9, 0, 9, 0, 2, 4, 3, 3, 3, 8, 3, 3, 8, 5, 7, 2, 2, 9, 1, 3, 6, 5, 4, 7, 1, 6, 0, 5, 8, 5, 2, 5, 4, 6, 7, 5, 9, 4, 4, 4
Offset: 0
Examples
0.6727383092174097953276872030889868689708768294102327312357145188219...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
- Robert E. Dressler, An elementary proof of a theorem of Erdös on the sum of divisors function, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.
- Paul Erdős, Some remarks on Euler's phi function and some related problems, Bulletin of the American Mathematical Society, Vol. 51, No. 8 (1945), pp. 540-544.
- Leonid G. Fel, Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization, arXiv:1108.0957 [math.NT], 2011, p. 6.
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y1).
- V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164, equation (4.2)-(4.3) on p. 162.
- László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv:1608.00795 [math.NT], 2016.
- László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.3, p. 18).
Programs
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Mathematica
$MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(p - 1)*Sum[1/(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
Equals Product_{p prime} (1 - 1/p) * Sum_{k>=0} 1/sigma(p^k) = Product_{p prime} ((p - 1)^2/p) * Sum_{k>=1} 1/(p^k - 1) = Product_{p prime} 1 - ((p - 1)^2/p) * Sum_{k>=1} 1/((p^k - 1)*(p^(k+1) - 1)).
Equals lim_{n->oo} (1/log(n))*Sum_{k=1..n} 1/sigma(k).
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} k/sigma(k) (the asymptotic mean of k/sigma(k)). - Amiram Eldar, Dec 23 2020
Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327. - Vaclav Kotesovec, Jun 14 2021
Extensions
More digits from Vaclav Kotesovec, Jun 13 2021
Comments