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A362412 The number of prime factors of the square root of the largest square dividing n, counted with multiplicity.

%I A362412 #40 May 07 2025 10:45:26
%S A362412 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,2,0,0,
%T A362412 0,2,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,3,0,0,0,1,
%U A362412 0,0,0,2,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0
%N A362412 The number of prime factors of the square root of the largest square dividing n, counted with multiplicity.
%C A362412 First differs from A366073 at n = 64.
%H A362412 Amiram Eldar, <a href="/A362412/b362412.txt">Table of n, a(n) for n = 1..10000</a>
%F A362412 a(n) = A001222(A000188(n)).
%F A362412 a(n) = A001222(A008833(n))/2.
%F A362412 Additive with a(p^e) = floor(e/2) = A004526(e).
%F A362412 a(n) >= 0, with equality if and only if n is squarefree (A005117).
%F A362412 a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).
%F A362412 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).
%F A362412 a(n) = (A001222(n) - A162642(n))/2. - _Ridouane Oudra_, Apr 19 2025
%t A362412 f[p_, e_] := Floor[e/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A362412 (PARI) a(n) = vecsum(factor(n)[, 2]\2);
%Y A362412 Cf. A000188, A000290, A001222, A004526, A005117, A008833, A154945, A162642, A349326, A366073.
%K A362412 nonn,easy
%O A362412 1,16
%A A362412 _Amiram Eldar_, Sep 28 2023