A307960 Decimal expansion of the asymptotic density of the coreful perfect numbers (A307958) that are generated from even primitives (A307959).

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

Offset

-2,1

Comments

Since the coreful perfect numbers are analogous to e-perfect numbers (A054979), the result of Hagis (see the formula and compare to A318645) can be also applied here.

If there is no odd perfect number, then this constant is the asymptotic density of all the coreful perfect numbers.

Links

Amiram Eldar, Table of n, a(n) for n = -2..10000

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

Formula

Equals Sum_{j>=1} beta(c(j))/c(j), where beta(k) = (6/Pi^2)*Product_{p|k}(p/(p+1)) and c(j) is the j-th even term of A307959.

Example

0.0093610474590681656384516304578441185615528428782...

Mathematica

f[p_] := 1/(3 * (2^p-1) * 2^(2*p-1)); v = MersennePrimeExponent/@Range[25]; RealDigits[(6/Pi^2)*Total[f/@v], 10, 100][[1]]

Crossrefs

Cf. A307958, A307959, A318645.

Sequence in context: A153618 A171051 A230158 * A224235 A363716 A169849

Adjacent sequences: A307957 A307958 A307959 * A307961 A307962 A307963

Keyword

nonn,cons

Author

Amiram Eldar, May 08 2019

Status

approved