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A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

%I A318366 #25 Jan 19 2019 03:41:26
%S A318366 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,
%T A318366 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,
%U A318366 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
%N A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).
%C A318366 Dirichlet convolution of A001222 with itself.
%H A318366 Antti Karttunen, <a href="/A318366/b318366.txt">Table of n, a(n) for n = 1..20160</a>
%H A318366 Antti Karttunen, <a href="/A318366/a318366.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A318366 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A318366 a(A025487(n)) = A322375(n). - _David A. Corneth_, Jan 12 2019
%F A318366 From _Robert Israel_, Jan 17 2019: (Start)
%F A318366 If x and y are coprime, a(x*y) = a(x)*A000005(y) + A000005(x)*a(y) + A000005(x*y)*A001222(x)*A001222(y).
%F A318366 If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)
%e A318366 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
%p A318366 f:= proc(n) local F,G,t,x;
%p A318366    F:= map(t -> t[2], ifactors(n)[2]);
%p A318366    G:= unapply(normal(mul((1-x^(t+1))/(1-x), t = F)),x);
%p A318366   (convert(F,`+`)-1)*D(G)(1) - (D@@2)(G)(1);
%p A318366 end proc:
%p A318366 map(f, [$1..100]); # _Robert Israel_, Jan 17 2019
%t A318366 Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]
%o A318366 (PARI) a(n) = sumdiv(n, d, bigomega(d)*bigomega(n/d)); \\ _Michel Marcus_, Aug 25 2018
%o A318366 (PARI) a(n) = bn = bigomega(n); sumdiv(n, d, bd = bigomega(d); bd * (bn - bd)) \\ _David A. Corneth_, Jan 12 2019
%Y A318366 Cf. A000005, A001222, A008578 (positions of 0's), A069264, A070288, A112967, A317938, A322375.
%K A318366 nonn
%O A318366 1,6
%A A318366 _Ilya Gutkovskiy_, Aug 24 2018