A370746 Decimal expansion of Sum_{k>=1} 1/(k*phi(2*k)), where phi is the Euler totient function (A000010).

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, 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, 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

Offset: 1

Author

Amiram Eldar, Feb 29 2024

Keywords

• nonn
• cons

Comments

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).

The constant appears in the asymptotic formula for the count of terms of A370745.

Examples

			1.76308527715028783029826265318407173005373855503028...
		

Links

• D. R. Heath-Brown, Cheryl E. Praeger and Aner Shalev, Permutation groups, simple groups, and sieve methods, Isr. J. Math., Vol. 148 (2005), pp. 347-375; alternative link.
• Cheryl E. Praeger, Using the finite simple groups, Gazette of the Australian Mathematical Society, Vol. 38, No. 2 (2011), pp. 93-97.
• Cheryl E. Praeger, Using the finite simple groups, Eureka, Vol. 61 (2011), pp. 64-67 (reprint of the Gazette paper).
• Cheryl E. Praeger, Using the finite simple groups, Asia Pacific Mathematics Newsletter, Vol. 1, No. 3 (2011), pp. 7-10 (reprint of the Gazette paper).

Crossrefs

Cf. A000010, A065484, A370745.

Programs

Mathematica

$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]]

PARI

(4/5)* prodeulerrat(1 + p/((p-1)^2*(p+1)))

Formula

Equals (4/5)* Product_{p prime} (1 + p/((p-1)^2*(p+1))) = (4/5) * A065484.