login
A362412
The number of primes factors of the square root of the largest square dividing n, counted with multiplicity.
1
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, 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, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0
OFFSET
1,16
LINKS
FORMULA
a(n) = A001222(A000188(n)).
a(n) = A001222(A008833(n))/2.
Additive with a(p^e) = floor(e/2) = A004526(e).
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).
MATHEMATICA
f[p_, e_] := Floor[e/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(factor(n)[, 2]\2);
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 28 2023
STATUS
approved