A361059 - OEIS

%I #6 Mar 01 2023 02:38:39

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

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

%U 3,3,7,9,7,4,9,3,5,4,6,4,9,0,6,6,6,5,1,3,9,2,1,7,9,2,9,5,4,7,3,9,6,2,5,7,3

%N Decimal expansion of the asymptotic mean of A000005(k)/A286324(k), the ratio between the number of divisors and the number of bi-unitary divisors.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiunitaryDivisor.html">Biunitary Divisor</a>.

%F Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A286324(k).

%F Equals Product_{p prime} (1 - (p-1)*log(1 - 1/p^2)/(2*p)).

%e 1.158854572650312100164480196393175149039104318857395...

%t $MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)*Log[1 - 1/p^2]/(2*p); 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, 106][[1]]

%Y Cf. A000005, A286324, A361060 (mean of the inverse ratio).

%Y Cf. A307869 (unitary analog), A308043.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Mar 01 2023