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A307960
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The decimal expansion of the asymptotic density of the coreful perfect numbers (A307958) that are generated from even primitives (A307959).
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3
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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
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OFFSET
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-2,1
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COMMENTS
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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 applied also here.
If there is no odd perfect number, then this constant is the asymptotic density of all the coreful perfect numbers.
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LINKS
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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.
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FORMULA
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Equals Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2)*Product_{p|n}(p/(p+1)) and c(i) are the even terms of A307959.
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EXAMPLE
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0.0093610474590681656384516304578441185615528428782...
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MATHEMATICA
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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]]
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CROSSREFS
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Cf. A307958, A307959, A318645.
Sequence in context: A153618 A171051 A230158 * A224235 A169849 A182547
Adjacent sequences: A307957 A307958 A307959 * A307961 A307962 A307963
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KEYWORD
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nonn,cons
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AUTHOR
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Amiram Eldar, May 08 2019
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STATUS
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approved
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