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%I #5 Feb 29 2024 06:20:38
%S 1,7,6,3,0,8,5,2,7,7,1,5,0,2,8,7,8,3,0,2,9,8,2,6,2,6,5,3,1,8,4,0,7,1,
%T 7,3,0,0,5,3,7,3,8,5,5,5,0,3,0,2,8,6,9,0,7,3,3,6,3,9,6,4,3,5,8,9,7,3,
%U 3,5,0,9,4,4,9,4,8,2,1,5,6,3,9,8,0,5,8,1,2,8,3,3,5,2,1,1,1,6,5,0,0,2,9,1,0
%N Decimal expansion of Sum_{k>=1} 1/(k*phi(2*k)), where phi is the Euler totient function (A000010).
%C The constant h in Heath-Brown et al. (2005). The asymptotic number of integers n below x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of A_{n-1} in A_n (where A_n is the simple alternating group), is ~ h*x/log(x).
%C The constant appears in the asymptotic formula for the count of terms of A370745.
%H D. R. Heath-Brown, Cheryl E. Praeger and Aner Shalev, <a href="https://doi.org/10.1007/BF02775443">Permutation groups, simple groups, and sieve methods</a>, Isr. J. Math., Vol. 148 (2005), pp. 347-375; <a href="https://core.ac.uk/download/pdf/96673.pdf">alternative link</a>.
%H Cheryl E. Praeger, <a href="https://www.austms.org.au/wp-content/uploads/Gazette/2011/May11/38(2)Web.pdf#page=29">Using the finite simple groups</a>, Gazette of the Australian Mathematical Society, Vol. 38, No. 2 (2011), pp. 93-97.
%H Cheryl E. Praeger, <a href="https://archim.org.uk/eureka/archive/Eureka-61.pdf">Using the finite simple groups</a>, Eureka, Vol. 61 (2011), pp. 64-67 (reprint of the Gazette paper).
%H Cheryl E. Praeger, <a href="http://www.asiapacific-mathnews.com/01/0103/0007_0010.pdf">Using the finite simple groups</a>, Asia Pacific Mathematics Newsletter, Vol. 1, No. 3 (2011), pp. 7-10 (reprint of the Gazette paper).
%F Equals (4/5)* Product_{p prime} (1 + p/((p-1)^2*(p+1))) = (4/5) * A065484.
%e 1.76308527715028783029826265318407173005373855503028...
%t $MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{1, 1, -2, 0, 1}, {0, 2, 3, 6, 5}, m]; RealDigits[(4/5)*Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
%o (PARI) (4/5)* prodeulerrat(1 + p/((p-1)^2*(p+1)))
%Y Cf. A000010, A065484, A370745.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, Feb 29 2024
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