A327576 - OEIS

%I #9 May 07 2021 08:34:47

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

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

%U 7,5,3,3,4,4,7,4,9,2,5,0,7,5,7,9,0,5,6

%N Decimal expansion of the constant that appears in the asymptotic formula for average order of the number of  infinitary divisors function (A037445).

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

%H Graeme L. Cohen and Peter Hagis, Jr., <a href="http://dx.doi.org/10.1155/S0161171293000456">Arithmetic functions associated with infinitary divisors of an integer</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

%F Equals Limit_{n->oo} A327573(n)/(2 * n * log(n)). [Corrected by _Amiram Eldar_, May 07 2021]

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

%e 0.366625276945381909556532720687001563033612155971009...

%t m = 1000; em = 10; f[x_] := Sum[Log[1 - 1/(1 + 1/x^(2^e))^2], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; $MaxExtraPrecision = 1500; RealDigits[(1/2)*Exp[f[1/2] + f[1/3]]* Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - (1/2)^k - (1/3)^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

%Y Cf. A037445, A050376, A327573.

%Y Cf. A059956 (corresponding constant for unitary divisors), A306071 (bi-unitary).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Sep 17 2019