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A308053
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Coreful abundant numbers: numbers k such that csigma(k) > 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).
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14
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72, 108, 144, 200, 216, 288, 324, 360, 400, 432, 504, 540, 576, 600, 648, 720, 756, 784, 792, 800, 864, 900, 936, 972, 1000, 1008, 1080, 1152, 1188, 1200, 1224, 1296, 1368, 1400, 1404, 1440, 1512, 1568, 1584, 1600, 1620, 1656, 1728, 1764, 1800, 1836, 1872, 1936
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OFFSET
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1,1
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COMMENTS
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The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 0.0262215..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022
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LINKS
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EXAMPLE
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72 is in the sequence since its coreful divisors are 6, 12, 18, 24, 36, 72, whose sum is 168 > 2 * 72.
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MATHEMATICA
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f[p_, e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] > 2n, AppendTo[s, n]], {n, 1, 2000}]; s
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PROG
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(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
(PARI) isok(k) = {my(f=factor(k)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1)-1) / (f[i, 1]-1)-1) > 2*k}; \\ Amiram Eldar, Sep 02 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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