A322887 Decimal expansion of the asymptotic mean value of the exponential abundancy index A051377(k)/k.

1, 1, 3, 6, 5, 7, 0, 9, 8, 7, 4, 9, 3, 6, 1, 3, 9, 0, 8, 6, 5, 2, 0, 7, 3, 1, 5, 2, 3, 8, 3, 8, 3, 2, 5, 9, 3, 4, 4, 8, 8, 0, 9, 0, 1, 8, 6, 3, 9, 5, 7, 2, 7, 6, 7, 8, 9, 0, 5, 2, 6, 5, 4, 4, 3, 1, 6, 2, 3, 9, 7, 2, 0, 3, 1, 5, 1, 5, 2, 8, 8, 3, 6, 8, 7, 6, 1, 3, 9, 2, 7, 2, 7, 4, 8, 9, 8, 5, 5, 2, 6, 2, 1, 9, 2

Offset: 1

Author

Amiram Eldar, Dec 29 2018

Keywords

• nonn
• cons

Examples

			1.13657098749361390865207315238383259344880901863957...
		

Links

• Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.

Crossrefs

Cf. A013661 (all divisors), A051377.

Programs

PARI

default(realprecision, 120); default(parisize, 2000000000);
my(kmax = 135); prodeulerrat(1 + (1 - 1/p) * sum(k = 1, kmax, 1/(p^(3*k)-p^k))) \\ Amiram Eldar, Mar 09 2024 (The calculation takes a few minutes.)

Formula

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} esigma(k)/k, where esigma(k) is the sum of exponential divisors of k (A051377).

Equals Product_{p prime} (1 + (1 - 1/p) * Sum_{k>=1} 1/(p^(3*k)-p^k)).

Extensions

a(7)-a(22) from Jon E. Schoenfield, Dec 30 2018

More terms from Amiram Eldar, Mar 09 2024