A088453 - OEIS

A088453

Decimal expansion of 1/zeta(3).

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

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OFFSET

0,1

COMMENTS

This is the probability that three randomly chosen integers are relatively prime (see A018805). - Gary McGuire, Dec 13 2004

This is also the probability that a random integer is cubefree. - Eugene Salamin, Dec 13 2004

On the other hand, the probability that three randomly-chosen integers are pairwise relatively prime is given by A065473. - Charles R Greathouse IV, Nov 14 2011

This is also the 'probability' that a random algebraic number's denominator is equal to its leading coefficient, see Arno, Robinson, & Wheeler. - Charles R Greathouse IV, Nov 12 2014

This is the probability that a random point on a cubic lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020

REFERENCES

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

LINKS

Table of n, a(n) for n=0..101.

Steven Arno, M. L. Robinson, and Ferell S. Wheeler, On denominators of algebraic numbers and integer polynomials, Journal of Number Theory 57:2 (April 1996), pp. 292-302.

Eric Weisstein's World of Mathematics, Relatively Prime.

FORMULA

From Amiram Eldar, Aug 20 2020: (Start)

Equals Sum_{k>=1} mu(k)/k^3, where mu is the Möbius function (A008683).

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

EXAMPLE

0.831907372580707468683126278821530734417...

MATHEMATICA

RealDigits[1/Zeta[3], 10, 120][[1]] (* Harvey P. Dale, May 31 2019 *)

PROG

(Maxima) fpprec : 200$ bfloat( 1/zeta(3))$ bfloat(%); /* Martin Ettl, Oct 15 2012 */

(PARI) 1/zeta(3) \\ Charles R Greathouse IV, Nov 12 2014