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A318366
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a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).
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5
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0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 10, 0, 8, 0, 8, 2, 2, 0, 20, 1, 2, 4, 8, 0, 12, 0, 20, 2, 2, 2, 24, 0, 2, 2, 20, 0, 12, 0, 8, 8, 2, 0, 40, 1, 8, 2, 8, 0, 20, 2, 20, 2, 2, 0, 34, 0, 2, 8, 35, 2, 12, 0, 8, 2, 12, 0, 52, 0, 2, 8, 8, 2, 12, 0, 40, 10, 2, 0, 34, 2, 2, 2, 20, 0, 34, 2, 8, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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Dirichlet convolution of A001222 with itself.
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LINKS
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FORMULA
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If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)
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EXAMPLE
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24 has 8 divisors, namely 1, 2, 3, 4, 6, 8, 12, 24, and four prime factors counted with multiplicity. The divisors have 0, 1, 1, 2, 2, 3, 3, 4 divisors respectively. So a(24) = 0 * (4 - 0) + 1 * (4 - 1) + 1 * (4 - 1) + 2 * (4 - 2) + 2 * (4 - 2) + 3 * (4 - 3) + 4 * (4 - 4) = 0 + 3 + 3 + 4 + 4 + 3 + 3 + 0 = 20. - David A. Corneth, Jan 12 2019
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MAPLE
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f:= proc(n) local F, G, t, x;
F:= map(t -> t[2], ifactors(n)[2]);
G:= unapply(normal(mul((1-x^(t+1))/(1-x), t = F)), x);
(convert(F, `+`)-1)*D(G)(1) - (D@@2)(G)(1);
end proc:
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MATHEMATICA
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Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]
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PROG
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(PARI) a(n) = sumdiv(n, d, bigomega(d)*bigomega(n/d)); \\ Michel Marcus, Aug 25 2018
(PARI) a(n) = bn = bigomega(n); sumdiv(n, d, bd = bigomega(d); bd * (bn - bd)) \\ David A. Corneth, Jan 12 2019
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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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