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A157294 Decimal expansion of 1575/Pi^6.

%I A157294 #13 May 12 2023 04:24:26
%S A157294 1,6,3,8,2,5,4,3,2,0,4,4,0,9,6,7,3,6,6,3,4,1,4,9,4,2,7,4,9,8,9,8,7,3,
%T A157294 5,5,4,9,1,8,7,0,2,5,2,6,6,4,4,3,4,4,7,1,8,0,7,2,9,0,0,6,7,4,8,9,2,5,
%U A157294 0,4,2,3,5,5,7,4,4,7,9,0,4,1,3,4,8,3,1,5,9,2,4,6,3,0,4,9,2,3,6,9,2,5,6,9,1
%N A157294 Decimal expansion of 1575/Pi^6.
%C A157294 Equals the asymptotic mean of the abundancy index of the 7-free numbers (numbers that are not divisible by a 7th power other than 1) (Jakimczuk and Lalín, 2022). - _Amiram Eldar_, May 12 2023
%H A157294 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 A157294 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A157294 Equals Product_{p = primes = A000040} (1+1/p^2+1/p^4+1/p^6).
%F A157294 Equals A013661/A013666 = A082020*A157290 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)) = 1575*A092746.
%e A157294 1.63825432044096736634149427498... = (1+1/2^2+1/2^4+1/2^6)*(1+1/3^2+1/3^4+1/3^6)*(1+1/5^2+1/5^4+1/5^6)*(1+1/7^2+1/7^4+1/7^6)*...
%p A157294 evalf(1575/Pi^6) ;
%t A157294 RealDigits[1575/Pi^6,10,120][[1]] (* _Harvey P. Dale_, May 26 2019 *)
%o A157294 (PARI) 1575/Pi^6 \\ _Charles R Greathouse IV_, Oct 01 2022
%Y A157294 Cf. A001248, A030514, A030516, A082020, A013661, A013666, A092746, A157290.
%K A157294 cons,easy,nonn
%O A157294 1,2
%A A157294 _R. J. Mathar_, Feb 26 2009