A328017 Decimal expansion of Product_{p prime} (1 - 7/p^3).

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

Offset

-1,1

Comments

This is the probability that k, k+1, ... k+6 all are cubefree, or equivalently, the asymptotic density of A328016.

Links

Table of n, a(n) for n=-1..104.

Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Example

0.084543731398668546996036273756803789380467263312806...

Mathematica

$MaxExtraPrecision = 600; m = 600; c = LinearRecurrence[{0, 0, 7}, {0, 0, -7}, m]; RealDigits[(1/8) * Exp[3*NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Prog

(PARI) prodeulerrat(1 - 7/p^3) \\ Amiram Eldar, Mar 17 2021

Crossrefs

Cf. A206256, A328016.

Sequence in context: A226581 A093821 A197256 * A195431 A273635 A299618

Adjacent sequences: A328014 A328015 A328016 * A328018 A328019 A328020

Keyword

nonn,cons

Author

Amiram Eldar, Oct 01 2019

Extensions

More terms from Vaclav Kotesovec, May 29 2020

Status

approved