A366078 -id:A366078 - cp's OEIS frontend

A366077 The number of prime factors of the cubefree part of n (A360539), counted with multiplicity.

0, 1, 1, 2, 1, 2, 1, 0, 2, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 1, 2, 2, 0, 3, 1, 3, 1, 0, 2, 2, 2, 4, 1, 2, 2, 1, 1, 3, 1, 3, 3, 2, 1, 1, 2, 3, 2, 3, 1, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Sep 28 2023

The sum of exponents smaller than 3 in the prime factorization of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Cf. A001222, A004709, A005117, A036966, A056169, A077761, A085541, A085548, A360539, A366076, A366078.

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

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

a(n) = A001222(A360539(n)).
a(n) = A001222(n) - A366076(n).
Additive with a(p^e) = e if e <= 2, and a(p^e) = 0 for e >= 3.
a(n) >= 0, with equality if and only if n is cubefull (A036966).
a(n) <= A001222(n), with equality if and only if n is cubefree (A004709).
a(n) >= A366078(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..m} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-2)/p^3 = A085548 - 2 * A085541 = 0.10272214144217842566... .

A366147 The number of divisors of the cubefree part of n (A360539).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 1, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 2, 3, 4, 1, 6, 2, 8, 2, 1, 4, 4, 4, 9, 2, 4, 4, 2, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 2, 4, 2, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 6, 4, 8, 2, 3, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 01 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A004709, A036966, A056671, A360539, A366077, A366078, A366145, A366148.

f[p_, e_] := If[e < 3, e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x < 3, x+1, 1), factor(n)[, 2]));

a(n) = A000005(A360539(n)).
a(n) = A000005(n)/A366145(n).
Multiplicative with a(p^e) = e+1 if e <= 2, and 1 otherwise.
a(n) >= 1, with equality if and only if n is cubefull (A036966).
a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 2/p^(3*s)).
From Vaclav Kotesovec, Oct 01 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 3/p^3 + 2/p^4) = 0.66219033176371496870504912254207846719824904470940603905284774924086...,
f'(1) = f(1) * Sum_{p prime} (9*p - 8) * log(p) / (p^4 - 3*p + 2) = f(1) * 1.04316863044761953555286128194165251303791613504188623828521117799260...
and gamma is the Euler-Mascheroni constant A001620. (End)

A366148 The sum of divisors of the cubefree part of n (A360539).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96

Offset: 1

Amiram Eldar, Oct 01 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A004709, A036966, A092261, A360539, A366077, A366078, A366146, A366147.

f[p_, e_] := If[e < 3, (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1));}

a(n) = A000203(A360539(n)).
a(n) = A000203(n)/A366146(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e <= 2, and 1 otherwise.
a(n) >= 1, with equality if and only if n is cubefull (A036966).
a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(3*s-2) - 1/p^(3*s-1)). CHECK
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p/(p+1) + 1/p - 1/p^4) = 0.63884633697952950095... .

cp's OEIS Frontend

A366077 The number of prime factors of the cubefree part of n (A360539), counted with multiplicity.

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A366147 The number of divisors of the cubefree part of n (A360539).

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A366148 The sum of divisors of the cubefree part of n (A360539).

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