A327574 Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).

7, 3, 0, 7, 1, 8, 2, 4, 2, 1, 2, 7, 3, 8, 4, 2, 2, 5, 8, 3, 8, 9, 7, 5, 4, 6, 8, 1, 7, 3, 5, 3, 0, 1, 6, 1, 9, 5, 7, 2, 5, 6, 4, 3, 3, 8, 6, 1, 7, 2, 7, 8, 6, 9, 7, 0, 7, 3, 3, 6, 7, 6, 2, 3, 0, 1, 0, 7, 9, 8, 8, 3, 3, 2, 8, 0, 0, 5, 3, 4, 6, 3, 7, 0, 2, 9, 9

Offset: 0

Author

Amiram Eldar, Sep 17 2019

Keywords

• nonn
• cons

Comments

The asymptotic mean of the infinitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A049417(k)/k = 1.461436... is twice this constant. - Amiram Eldar, Jun 13 2020

Examples

			0.730718242127384225838975468173530161957256433861727...
		

References

• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

Links

• Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with infinitary divisors of an integer, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

Crossrefs

Cf. A049417, A050376, A327566.

Cf. A013661 (corresponding constant for all divisors), A275480 (exponential), A306633 (unitary), A307160 (bi-unitary).

Programs

Mathematica

$MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Sum[Log[1 + x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals Limit_{k->oo} A327566(k)/k^2.

Equals (1/2) * Product_{P} (1 + 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).