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A307960
Decimal expansion of the asymptotic density of the coreful perfect numbers (A307958) that are generated from even primitives (A307959).
3
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
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 08 2019
STATUS
approved