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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).
4
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
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 10 2019
EXTENSIONS
More digits from Vaclav Kotesovec, Jun 13 2021
STATUS
approved