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

Author

N. J. A. Sloane, May 09 2003

Keywords

• nonn
• cons

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

Examples

			1.51981775463506657...
		

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.

Crossrefs

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

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // G. C. Greubel, Oct 18 2019

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 *)

PARI

15/Pi^2 \\ Michel Marcus, Oct 18 2019

Sage

numerical_approx(15/pi^2, digits=100) # G. C. Greubel, Oct 18 2019

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