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%I #11 May 12 2023 04:23:53
%S 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,
%T 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,
%U 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
%N Decimal expansion of 31185/(2*Pi^8).
%C 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
%H 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 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F 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).
%F Equals A013661/A013668 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)+1/A030514(i)^2) = 15592.5*A092748.
%e 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)*...
%p evalf(31185/2/Pi^8) ;
%t RealDigits[31185/(2*Pi^8),10,120][[1]] (* _Harvey P. Dale_, Mar 30 2018 *)
%o (PARI) 31185/2/Pi^8 \\ _Charles R Greathouse IV_, Oct 01 2022
%Y Cf. A001248, A030514, A030514, A030516, A082020, A013661, A013668, A092748.
%K cons,easy,nonn
%O 1,2
%A _R. J. Mathar_, Feb 26 2009
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