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A361793
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Sum of the squares d^2 of the divisors d satisfying d^3|n.
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1
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1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 10, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5
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OFFSET
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1,8
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COMMENTS
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The Mobius transform is 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, ... = n^(2/3)*A010057(n).
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LINKS
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FORMULA
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a(n) = Sum_{d^3|n} d^2.
Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - Amiram Eldar, Mar 24 2023
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MAPLE
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gsigma := proc(n, z, k)
local a, d ;
a := 0 ;
for d in numtheory[divisors](n) do
if modp(n, d^k) = 0 then
a := a+d^z ;
end if ;
end do:
a ;
end proc:
seq( gsigma(n, 2, 3), n=1..80) ;
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MATHEMATICA
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f[p_, e_] := (p^(2*(Floor[e/3] + 1)) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ Michel Marcus, Mar 24 2023
(Python)
from math import prod
from sympy import factorint
def A361793(n): return prod((p**(e//3+1<<1)-1)//(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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