A366076 - OEIS

%I #8 Sep 29 2023 04:10:08

%S 0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,5,0,0,

%T 0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,6,0,0,0,0,

%U 0,0,0,3,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0

%N The number of primes factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

%C The sum of exponents larger than 2 in the prime factorization of n.

%C The number of distinct primes factors of the largest divisor of n that is a cubefull number is A295659(n).

%H Amiram Eldar, <a href="/A366076/b366076.txt">Table of n, a(n) for n = 1..10000</a>

%H Rafael Jakimczuk and Matilde Lalín, <a href="http://math.colgate.edu/~integers/w113/w113.pdf">Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences</a>, Integers, Vol. 22 (2022), Article #A113.

%F a(n) = A001222(A360540(n)).

%F a(n) = A001222(n) - A366077(n).

%F Additive with a(p^e) = 0 if e <= 2, and a(p^e) = e for e >= 3.

%F a(n) >= 0, with equality if and only if n is cubefree (A004709).

%F a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).

%F a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2/p^3 + 1/(p^2*(p-1))) = 2 * A085541 + A152441 = 0.67043452760761670220... .

%t f[p_, e_] := If[e < 3, 0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecsum(apply(x -> if(x < 3, 0, x), factor(n)[, 2]));

%Y Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

%Y Similar sequence: A275812 (number of primes factors of the powerful part).

%K nonn,easy

%O 1,8

%A _Amiram Eldar_, Sep 28 2023