A318366 - OEIS

A318366

a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

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

1,6

COMMENTS

Dirichlet convolution of A001222 with itself.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20160

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(A025487(n)) = A322375(n). - David A. Corneth, Jan 12 2019

From Robert Israel, Jan 17 2019: (Start)

If x and y are coprime, a(x*y) = a(x)*A000005(y) + A000005(x)*a(y) + A000005(x*y)*A001222(x)*A001222(y).

If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)

EXAMPLE

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

MAPLE

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:

map(f, [$1..100]); # Robert Israel, Jan 17 2019

MATHEMATICA

Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]

PROG

(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