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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

%I #20 Jul 02 2023 03:18:33

%S 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,

%T 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,

%U 1,8,4,1,5,4,6,2,4,9,0,8,6,4,7,8,1,5,7

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

%C 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.

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

%H Amiram Eldar, <a href="/A307960/b307960.txt">Table of n, a(n) for n = -2..10000</a>

%H Peter Hagis, <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 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.

%e 0.0093610474590681656384516304578441185615528428782...

%t 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]]

%Y Cf. A307958, A307959, A318645.

%K nonn,cons

%O -2,1

%A _Amiram Eldar_, May 08 2019

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Last modified September 9 07:24 EDT 2024. Contains 375762 sequences. (Running on oeis4.)