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A226602
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Number of ordered triples (i,j,k) with i*j*k = n, i,j,k >= 0 and gcd(i,j,k) <= 1.
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11
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1, 1, 3, 3, 6, 3, 9, 3, 9, 6, 9, 3, 18, 3, 9, 9, 12, 3, 18, 3, 18, 9, 9, 3, 27, 6, 9, 9, 18, 3, 27, 3, 15, 9, 9, 9, 36, 3, 9, 9, 27, 3, 27, 3, 18, 18, 9, 3, 36, 6, 18, 9, 18, 3, 27, 9, 27, 9, 9, 3, 54, 3, 9, 18, 18, 9, 27, 3, 18, 9, 27, 3, 54, 3, 9, 18, 18
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OFFSET
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0,3
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COMMENTS
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Note that gcd(0,m) = m for any m.
a(n) is the number of cubefree divisors summed over the divisors of n. In other words, a(n) = Sum_{d|n} A073184(d). - Geoffrey Critzer, Mar 20 2015
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LINKS
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FORMULA
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If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.
Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)
Multiplicative with a(p^e) = 3*e, p prime and e>0.
Dirichlet inverse b(n), n>0, is multiplicative with b(1) = 1, and for p prime and e>0: b(p^e)=0 if e mod 3 = 0 otherwise b(p^e)=3*(-1)^(e mod 3).
Sum_{k=1..n} a(k) ~ n/(2*Zeta(3)) * (log(n)^2 + 2*log(n) * (-1 + 3*gamma - 3*Zeta'(3)/Zeta(3)) + 2 + 6*gamma^2 - 6*sg1 + 6*Zeta'(3)/Zeta(3) + 18*Zeta'(3)^2/Zeta(3)^2 - 6*gamma*(1 + 3*Zeta'(3)/Zeta(3)) - 9*Zeta''(3)/Zeta(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 07 2019
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MAPLE
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with(numtheory):
b:= proc(n, t, g) option remember; `if`(t=0,
`if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
igcd(g, d)), d=divisors(n)))
end:
a:= n-> `if`(n=0, 1, b(n, 2, 0)):
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MATHEMATICA
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f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] <= 1, {i, j, k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; [Table[f[n], {n, 0, 100}]
a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* Amiram Eldar, Sep 14 2020 *)
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PROG
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(Python)
from math import prod
from sympy import factorint
def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # Chai Wah Wu, Dec 26 2022
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CROSSREFS
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Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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