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.

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, 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, 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

Offset: 0

Author

Amiram Eldar, Apr 15 2025

Keywords

• nonn
• cons

Comments

The asymptotic mean of the inverse ratio A034444(k)/A365498(k) is zeta(3)/zeta(6) (A157289).

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).

Examples

			0.91429441180198062448296176452156718437854669178193...
		

Crossrefs

The unitary analog of A361062.

Cf. A034444, A157289, A365498, A383057.

Programs

Mathematica

$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]]

PARI

prodeulerrat(1 - 1/(2*p^3))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A365498(k)/A034444(k).

Equals Product_{p prime} (1 - 1/(2*p^3)).

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)).