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A329484
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Dirichlet convolution of the Louiville function with itself.
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1
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1, -2, -2, 3, -2, 4, -2, -4, 3, 4, -2, -6, -2, 4, 4, 5, -2, -6, -2, -6, 4, 4, -2, 8, 3, 4, -4, -6, -2, -8, -2, -6, 4, 4, 4, 9, -2, 4, 4, 8, -2, -8, -2, -6, -6, 4, -2, -10, 3, -6, 4, -6, -2, 8, 4, 8, 4, 4, -2, 12, -2, 4, -6, 7, 4, -8, -2, -6, 4, -8, -2, -12, -2
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OFFSET
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1,2
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COMMENTS
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Up to sign this sequence partitions the positive integers in the same way asA008836.Additional interesting partitions exist when values of this sequence are taken into account.
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(2*s)^2/zeta(s)^2. -Amiram Eldar,Dec 05 2022
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MATHEMATICA
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a[n_]:= DivisorSum[n, LiouvilleLambda[#] * LiouvilleLambda[n/#] &]; Array[a, 100] (*Amiram Eldar,Jan 18 2020 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, (-1)^big Omega (d) * (-1)^big Omega (n/d))
(PARI) a(n) = {numdiv(n)*(-1)^big Omega (n)} \\Andrew Howroyd,Sep 15 2020
(Python)
from math import prod
from sympy import factorint
defA329484(n): return prod(-e-1 if e&1 else e+1 for e in factorint(n).values()) #Chai Wah Wu,Dec 23 2022
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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