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

0,1

Comments

Erdös proved the existence of this constant. Dressler found its explicit form.

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).

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

Example

0.6727383092174097953276872030889868689708768294102327312357145188219...

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]]

Crossrefs

Cf. A000203, A074753, A345327.

Sequence in context: A070059 A088667 A153971 * A316165 A267251 A259526

Adjacent sequences: A308036 A308037 A308038 * A308040 A308041 A308042

Keyword

nonn,cons

Author

Amiram Eldar, May 10 2019

Extensions

More digits from Vaclav Kotesovec, Jun 13 2021

Status

approved