A362412 - OEIS

%I #29 Sep 29 2023 04:08:45

%S 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 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 0,0,0,2,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0

%N The number of primes factors of the square root of the largest square dividing n, counted with multiplicity.

%H Amiram Eldar, <a href="/A362412/b362412.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001222(A000188(n)).

%F a(n) = A001222(A008833(n))/2.

%F Additive with a(p^e) = floor(e/2) = A004526(e).

%F a(n) >= 0, with equality if and only if n is squarefree (A005117).

%F a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).

%t f[p_, e_] := Floor[e/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecsum(factor(n)[, 2]\2);

%Y Cf. A000188, A000290, A001222, A004526, A005117, A008833, A154945, A349326.

%K nonn,easy

%O 1,16

%A _Amiram Eldar_, Sep 28 2023