A082020 Decimal expansion of 15/Pi^2.

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

Offset

1,2

Comments

3/(2*Pi^2) (the same decimal expansion with an offset 0) is the probability that the greatest common divisor of two numbers selected at random is 2 (Christopher, 1956). - Amiram Eldar, May 23 2020

Sum of 1/n^2 over all squarefree n, see Penn link. - Charles R Greathouse IV, Jan 01 2022

Equals the asymptotic mean of the abundancy index of the cubefree numbers (A004709) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.

Werner Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Vol. 14, No. 2 (2015), pp. 73-88.

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).

Michael Penn, Irregular numbers and the coolest sum I have ever seen!, 2021 video.

S. Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.

V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link (p. 230).

Eric Weisstein's World of Mathematics, Prime Sums.

Eric Weisstein's World of Mathematics, Moebius Function.

Eric Weisstein's World of Mathematics, Prime Products.

Index entries for transcendental numbers.

Formula

Product_{n >= 1} (1+1/prime(n)^2) = 15/Pi^2 (Ramanujan).

Equals zeta(2)/zeta(4) = A013661/A013662 = Sum_{n>=1} mu(n)^2/n^2 = Sum_{n>=1} |mu(n)|/n^2 . - Enrique Pérez Herrero, Jan 15 2012

Equals Sum_{n>=1} 1/A005117(n)^2 . - Enrique Pérez Herrero, Mar 30 2012

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} psi(k)/k, where psi(k) is the Dedekind psi function (A001615). - Amiram Eldar, May 12 2019.

Equals Sum_{k>=1} A007434(k)/k^4. - Amiram Eldar, Jan 25 2024

Example

1.51981775463506657...

Maple

evalf(15/Pi^2, 120); # G. C. Greubel, Oct 18 2019

Mathematica

A082020[digits_] := First[RealDigits[Zeta[2]/Zeta[4], 10, digits]]; A082020[100] (* Enrique Pérez Herrero, Jan 15 2012 *)
RealDigits[15/Pi^2, 10, 120][[1]] (* Harvey P. Dale, Jun 23 2019 *)

Prog

(PARI) 15/Pi^2 \\ Michel Marcus, Oct 18 2019
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // G. C. Greubel, Oct 18 2019
(Sage) numerical_approx(15/pi^2, digits=100) # G. C. Greubel, Oct 18 2019

Crossrefs

Cf. A001615, A004709, A005117, A007434, A013661, A013662, A157290.

Sequence in context: A154605 A114594 A021662 * A256559 A182498 A147406

Adjacent sequences: A082017 A082018 A082019 * A082021 A082022 A082023

Keyword

nonn,cons

Author

N. J. A. Sloane, May 09 2003

Status

approved