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A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

%I A366123 #9 Sep 30 2023 21:56:56
%S A366123 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,
%T A366123 0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,2,0,0,0,0,
%U A366123 0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0
%N A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.
%C A366123 First differs from A295659 at n = 64.
%C A366123 The number of distinct prime factors of the cube root of the largest cube dividing n is A295659(n).
%H A366123 Amiram Eldar, <a href="/A366123/b366123.txt">Table of n, a(n) for n = 1..10000</a>
%F A366123 a(n) = A001222(A053150(n)).
%F A366123 a(n) = A001222(A008834(n))/3.
%F A366123 Additive with a(p^e) = floor(e/3) = A002264(e).
%F A366123 a(n) >= 0, with equality if and only if n is cubefree (A004709).
%F A366123 a(n) <= A001222(n)/3, with equality if and only if n is a positive cube (A000578 \ {0}).
%F A366123 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^3-1) = 0.194118... (A286229).
%t A366123 f[p_, e_] := Floor[e/3]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366123 (PARI) a(n) = vecsum(apply(x -> x\3, factor(n)[, 2]));
%Y A366123 Cf. A000578, A001222, A002264, A008834, A004709, A053150, A286229, A295659.
%Y A366123 Cf. A061704 (number of divisors), A333843 (sum of divisors).
%K A366123 nonn,easy
%O A366123 1,64
%A A366123 _Amiram Eldar_, Sep 30 2023