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A349712
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a(n) = Sum_{d|n} sopf(d) * sopf(n/d).
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2
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0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110
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OFFSET
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1,4
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COMMENTS
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Dirichlet convolution of A008472 with itself.
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LINKS
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FORMULA
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Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021
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MATHEMATICA
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sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]
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PROG
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(PARI) sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021
(Python)
from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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