A286229 -id:A286229 - cp's OEIS frontend

A375145 Numbers whose prime factorization has exactly one exponent that is larger than 2.

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 343, 344

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A046099 and first differs from it at n = 35: A046099(35) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p^3-1) = A286229 / A002117 = 0.16148833663564192901... .

			8 = 2^3 is a term since its prime factorization has exactly one exponent, 3, that is larger than 2.

Subsequence of A046099, A356862.

Cf. A002117, A057521, A190641, A246549, A286229, A375146.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 2 &)] == 1; Select[Range[350], q]

is(k) = #select(x -> x > 2, factor(k)[, 2]) == 1;

A322664 a(n) = n^2 * Sum_{p^k|n} Sum_{j=1..k} 1/p^(2*j), where p are primes dividing n with multiplicity k.

Original entry on oeis.org

0, 1, 1, 5, 1, 13, 1, 21, 10, 29, 1, 61, 1, 53, 34, 85, 1, 121, 1, 141, 58, 125, 1, 253, 26, 173, 91, 261, 1, 361, 1, 341, 130, 293, 74, 565, 1, 365, 178, 589, 1, 673, 1, 621, 331, 533, 1, 1021, 50, 729, 298, 861, 1, 1093, 146, 1093, 370, 845, 1, 1669, 1, 965

Offset: 1

Daniel Suteu, Dec 22 2018

nonn

The generalized formula is f(n,m) = n^m * Sum_{p^k|n} Sum_{j=1..k} 1/p^(m*j), where f(n,0) = A001222(n) and f(n,1) = A095112(n).
From Ridouane Oudra, Jul 21 2025: (Start)
a(n) is the sum of (n/d)^2 over all prime powers d which divide n.
Using the previous generalized formula we have :
f(n,m) = Sum_{d|n, d is a prime power} (n/d)^m.
f(n,m) = Sum_{d|n} bigomega(d)*J_m(n/d), where J_m is the m-th Jordan totient function. (End)

			The prime factorization of 24 is 2^3 * 3, so a(24) = 24^2 * (1/2^2 + 1/2^(2*2) + 1/2^(2*3) + 1/3^2) = 253.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001222, A095112, A286229, A007434.

a(n) = my(f=factor(n)); sum(k=1, #f~, sum(j=1, f[k,2], n^2 / f[k,1]^(2*j)));

Sum_{k=1..n} a(k) ~ A286229 * A000330(n).

a(n) = Sum_{d|n} bigomega(d)*J_2(n/d), where J_2 = A007434. - Ridouane Oudra, Jul 21 2025

A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 30 2023

First differs from A295659 at n = 64.

The number of distinct prime factors of the cube root of the largest cube dividing n is A295659(n).

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

Cf. A000578, A001222, A002264, A008834, A004709, A053150, A286229, A295659.

Cf. A061704 (number of divisors), A333843 (sum of divisors).

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

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

a(n) = A001222(A053150(n)).
a(n) = A001222(A008834(n))/3.
Additive with a(p^e) = floor(e/3) = A002264(e).
a(n) >= 0, with equality if and only if n is cubefree (A004709).
a(n) <= A001222(n)/3, with equality if and only if n is a positive cube (A000578 \ {0}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^3-1) = 0.194118... (A286229).

cp's OEIS Frontend

A375145 Numbers whose prime factorization has exactly one exponent that is larger than 2.

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A322664 a(n) = n^2 * Sum_{p^k|n} Sum_{j=1..k} 1/p^(2*j), where p are primes dividing n with multiplicity k.

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A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

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Mathematica

PARI

Formula