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A180114
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a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).
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8
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1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 195, 121, 217, 280, 133, 156, 255, 403, 183, 399, 465, 600, 403, 364, 511, 819, 307, 847, 400, 381, 855, 961, 1240, 741, 931, 1092, 1023, 553, 1651, 781, 1815, 1240, 1281, 1093, 1767, 1953, 871, 2520, 2821, 993, 1995
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OFFSET
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1,2
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LINKS
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FORMULA
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Sum_{a(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)
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EXAMPLE
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Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.
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MAPLE
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emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2], L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L), sn]; print(n, ifactor(n), sn, ifactor(sn)) fi; od; od;
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MATHEMATICA
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pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* Giovanni Resta, Feb 06 2018 *)
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PROG
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(PARI) lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ Amiram Eldar, May 12 2023
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A180114_gen(): # generator of terms
for n in count(1):
f = factorint(n)
if all(e>1 for e in f.values()):
yield prod((p**(e+1)-1)//(p-1) for p, e in f.items())
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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