A322887 - OEIS

%I #23 Mar 09 2024 08:18:39

%S 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,

%T 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,

%U 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

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

%H Peter Hagis, Jr., <a href="http://dx.doi.org/10.1155/S0161171288000407">Some results concerning exponential divisors</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.

%F 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).

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

%e 1.13657098749361390865207315238383259344880901863957...

%o (PARI) default(realprecision, 120); default(parisize, 2000000000);

%o 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.)

%Y Cf. A013661 (all divisors), A051377.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Dec 29 2018

%E a(7)-a(22) from _Jon E. Schoenfield_, Dec 30 2018

%E More terms from _Amiram Eldar_, Mar 09 2024