A366076 -id:A366076 - 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... .

A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 01 2023

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

Cf. A000005, A036966, A357669, A360540, A366076, A366146, A366147.

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

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

a(n) = A000005(A360540(n)).
a(n) = A000005(n)/A366147(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000005(n), with equality if and only if n is cubefull (A036966).
Multiplicative with a(p^e) = 1 if e <= 2 and e+1 otherwise.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 3/p^(3*s) - 2/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 + 1/p^4 - 2/p^5) = 1.76434793373691907811... .

A366146 The sum of divisors of the largest divisor of n that is a cubefull number (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 40, 1, 1, 1, 1, 63, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 40, 1, 15, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1

Offset: 1

Amiram Eldar, Oct 01 2023

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

Cf. A000203, A036966, A295294, A360540, A366076, A366145, A366148.

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

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

a(n) = A000203(A360540(n)).
a(n) = A000203(n)/A366148(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000203(n), with equality if and only if n is cubefull (A036966).
Multiplicative with a(p^e) = 1 if e <= 2 and (p^(e+1)-1)/(p-1) otherwise.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-3) - 1/p^(4*s-2)).

cp's OEIS Frontend

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

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A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

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A366146 The sum of divisors of the largest divisor of n that is a cubefull number (A036966).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula