A157296 Decimal expansion of 31185/(2*Pi^8).

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

Offset: 1

Author

R. J. Mathar, Feb 26 2009

Keywords

• cons
• easy
• nonn

Comments

Equals the asymptotic mean of the abundancy index of the 9-free numbers (numbers that are not divisible by a 9th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

Examples

			1.64329968185709999227... = (1+1/2^2+1/2^4+1/2^6+1/2^8)*(1+1/3^2+1/3^4+1/3^6+1/3^8)*(1+1/5^2+1/5^4+1/5^6+1/5^8)*...
		

Links

• 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).
• Index entries for transcendental numbers.

Crossrefs

Cf. A001248, A030514, A030514, A030516, A082020, A013661, A013668, A092748.

Programs

Maple

evalf(31185/2/Pi^8) ;

Mathematica

RealDigits[31185/(2*Pi^8),10,120][[1]] (* Harvey P. Dale, Mar 30 2018 *)

PARI

31185/2/Pi^8 \\ Charles R Greathouse IV, Oct 01 2022

Formula

Equals Product_{p = primes = A000040} (1+1/p^2+1/p^4+1/p^6+1/p^8). The variant Product_{p} (1+1/p^2+1/p^6+1/p^8) equals A082020*Product_{p} (1+1/p^6) = A082020*zeta(6)/zeta(12) = 10135125/(691*Pi^8).

Equals A013661/A013668 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)+1/A030514(i)^2) = 15592.5*A092748.