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A157292 Decimal expansion of 315/(2*Pi^4).

%I A157292 #20 May 12 2023 04:25:28
%S A157292 1,6,1,6,8,9,2,2,0,5,1,1,2,7,8,2,7,9,2,2,9,1,5,6,3,3,6,4,5,7,1,1,9,4,
%T A157292 3,2,7,3,3,7,8,7,8,7,9,1,9,4,8,0,2,6,3,7,8,1,1,1,4,6,5,5,8,6,8,3,5,8,
%U A157292 5,1,8,7,1,3,9,9,4,2,7,4,3,9,2,2,8,9,0,0,1,5,3,9,0,0,8,2,5,2,2,6,3,6,2,7,2
%N A157292 Decimal expansion of 315/(2*Pi^4).
%C A157292 Equals the asymptotic mean of the abundancy index of the 5-free numbers (numbers that are not divisible by a 5th power other than 1) (Jakimczuk and Lalín, 2022). - _Amiram Eldar_, May 12 2023
%H A157292 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
%H A157292 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A157292 Equals Product_{p = primes} (1 + 1/p^2 + 1/p^4), whereas, the product over (1 + 2/p^2 + 1/p^4) equals A082020^2.
%F A157292 Equals A013661/A013664 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)).
%F A157292 Equals 315*A092744/2.
%F A157292 Equals Sum_{n>=1} 1/A004709(n)^2. - _Geoffrey Critzer_, Feb 16 2015
%e A157292 1.61689220511... = (1+1/2^2+1/2^4)*(1+1/3^2+1/3^4)*(1+1/5^2+1/5^4)*(1+1/7^2+1/7^4)*...
%p A157292 evalf(315/2/Pi^4) ;
%t A157292 RealDigits[N[Zeta[2]/Zeta[6], 150]][[1]] (* _Geoffrey Critzer_, Feb 16 2015 *)
%o A157292 (PARI) 315/2/Pi^4 \\ _Charles R Greathouse IV_, Oct 01 2022
%Y A157292 Cf. A001248, A004709, A030514, A092744, A013661, A013664, A082020.
%K A157292 cons,easy,nonn
%O A157292 1,2
%A A157292 _R. J. Mathar_, Feb 26 2009