cp's OEIS Frontend

A383058 Decimal expansion of the asymptotic mean of A365498(k)/A034444(k), the ratio between the number of cubefree unitary divisors and the number of unitary divisors over the positive integers.

%I A383058 #5 Apr 15 2025 10:26:04
%S A383058 9,1,4,2,9,4,4,1,1,8,0,1,9,8,0,6,2,4,4,8,2,9,6,1,7,6,4,5,2,1,5,6,7,1,
%T A383058 8,4,3,7,8,5,4,6,6,9,1,7,8,1,9,3,6,8,6,6,5,9,1,9,9,7,9,7,6,7,0,0,8,5,
%U A383058 3,4,3,8,8,3,2,0,5,6,7,6,0,8,0,0,7,1,0,7,6,7,3,6,5,0,0,4,2,6,2,6,0,5,8,2,4
%N A383058 Decimal expansion of the asymptotic mean of A365498(k)/A034444(k), the ratio between the number of cubefree unitary divisors and the number of unitary divisors over the positive integers.
%C A383058 The asymptotic mean of the inverse ratio A034444(k)/A365498(k) is zeta(3)/zeta(6) (A157289).
%C A383058 In general, the asymptotic mean of the inverse ratio, between the number of unitary divisors and the number of k-free (i.e., not divisible by a k-th power other than 1) unitary divisors over the positive integers, for k >= 2, is zeta(k)/zeta(2*k).
%F A383058 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A365498(k)/A034444(k).
%F A383058 Equals Product_{p prime} (1 - 1/(2*p^3)).
%F A383058 In general, the asymptotic mean of the ratio between the number of k-free unitary divisors and the number of unitary divisors over the positive integers, for k >= 2, is Product_{p prime} (1 - 1/(2*p^k)).
%e A383058 0.91429441180198062448296176452156718437854669178193...
%t A383058 $MaxExtraPrecision = 300; m = 300; f[p_] := 1 - 1/(2*p^3); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n]), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
%o A383058 (PARI) prodeulerrat(1 - 1/(2*p^3))
%Y A383058 The unitary analog of A361062.
%Y A383058 Cf. A034444, A157289, A365498, A383057.
%K A383058 nonn,cons
%O A383058 0,1
%A A383058 _Amiram Eldar_, Apr 15 2025