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A295294
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Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).
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8
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1, 1, 1, 7, 1, 1, 1, 15, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 15, 31, 1, 40, 7, 1, 1, 1, 63, 1, 1, 1, 91, 1, 1, 1, 15, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 40, 1, 15, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 195, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 15, 1, 13, 1, 7, 1, 1, 1, 63
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OFFSET
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1,4
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LINKS
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FORMULA
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Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.
a(n) = 1 iff n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 09 2023
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MATHEMATICA
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f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 08 2022 *)
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PROG
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(Scheme)
;; With memoization-macro definec:
(PARI) a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ Amiram Eldar, Oct 08 2022
(Python)
from math import prod
from sympy import factorint
def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # Chai Wah Wu, Nov 14 2022
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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