A361064 Multiplicative with a(p^e) = sigma_3(e), where sigma_3 = A001158.
1, 1, 1, 9, 1, 1, 1, 28, 9, 1, 1, 9, 1, 1, 1, 73, 1, 9, 1, 9, 1, 1, 1, 28, 9, 1, 28, 9, 1, 1, 1, 126, 1, 1, 1, 81, 1, 1, 1, 28, 1, 1, 1, 9, 9, 1, 1, 73, 9, 9, 1, 9, 1, 28, 1, 28, 1, 1, 1, 9, 1, 1, 9, 252, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 9, 9, 1, 1, 1, 73, 73, 1, 1, 9
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
g[p_, e_] := DivisorSigma[3, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
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PARI
a(n) = vecprod(apply(x -> sigma(x, 3), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025
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Python
from math import prod from sympy import factorint, divisor_sigma def A361064(n): return prod(divisor_sigma(e,3) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023
Formula
Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_3(e) / p^(e*s)).
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_3(e) - sigma_3(e-1)) / p^e) = 136.775196585091127831467103699999450735835551529525277016916082455332230986...