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Decimal expansion of the constant that appears in the asymptotic formula for average order of an infinitary analog of Euler's phi function (A091732).
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%I #5 Sep 17 2019 08:26:45

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

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

%U 2,4,0,0,7,7,3,7,9,6,8,7,4,8,8,3,9,9,7

%N Decimal expansion of the constant that appears in the asymptotic formula for average order of an infinitary analog of Euler's phi function (A091732).

%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_{k->oo} A327572(k)/k^2.

%F 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).

%e 0.328935838840335516355748487365220229577066523794694...

%t $MaxExtraPrecision = 1500; m = 1500; 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]]

%Y Cf. A091732, A050376, A327572.

%Y Cf. A104141 (corresponding constant for phi), A065463 (unitary), A306071 (bi-unitary).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Sep 17 2019