A375072 -id:A375072 - cp's OEIS frontend

A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A176297 and A375072, and first differs from them at n = 20: A176297(20) = A375072(20) = 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+p^2+p^3) = 0.089602607198058453295... .

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

Subsequence of A046100, A176297, A375072 and A375145.
Subsequences: A030078, A048109, A065036, A143610, A163569, A179670, A179695, A179700, A189975, A189984, A190109, A190378, A190382.
Cf. A002117.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, MemberQ[e, 3] && Count[e, _?(# < 3 &)] == Length[e] - 1]; Select[Range[600], q]

isok(k) = {my(e = factor(k)[, 2]~); select(x -> x > 2, e) == [3];}

A382966 The number of non-unitary prime divisors of the n-th biquadratefree number that is not cubefree.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Apr 10 2025

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

Cf. A002117, A013662, A056170, A375072, A376366, A382425, A382965, A382968.

f[k_] := If[k == 1, Nothing, Module[{e = FactorInteger[k][[;; , 2]]}, If[Max[e] == 3, Count[e, _?(# > 1 &)], Nothing]]]; Array[f, 1000]

list(lim) = {my(e); for(k = 2, lim, e = factor(k)[, 2]; if(vecmax(e) == 3, print1(#select(x -> x > 1, e), ", ")));}

a(n) = A056170(A375072(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((1/zeta(4)) * Sum_{p prime} (1/(p^2+1)) - (1/zeta(3)) * Sum_{p prime} ((p-1)/(p^3-1))) / (1/zeta(4) - 1/zeta(3)) = 1.20757893653588072073... .

A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A163569 at n = 25: a(25) = 2520 = 2^3 * 3^2 * 5 * 7 is not a term of A163569.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 2, 3}.
The asymptotic densities of this sequence and A375074 are equal (0.0156712..., see A375074 for a formula), since the terms in A375074 that are not in this sequence (A375073) have a density 0.

Intersection of A375072 and A317090.
Equals A375074 \ A375073.
Subsequence of A046100 and A176297.
A163569 is a subsequence.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A136568, A215267.

Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]

is(k) = Set(factor(k)[,2]) == [1, 2, 3];

A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 84, 88, 90, 92, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 180, 184

Offset: 1

Amiram Eldar, Apr 10 2025

Subsequence of A252849 and first differs from it at n = 22: A252849(22) = 64 = 2^6 is not a term of this sequence.
Subsequence of A375229 and differs from it by not having the terms 1, 256, 512, 768, 1024, ... .
Numbers whose prime factorization has least one exponent that equals 2 or 3 and no higher exponent.
Numbers k such that 2 <= A051903(k) <= 3.
The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(2) = A215267 - A059956 = 0.3160113... .

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

Disjoint union of A067259 and A375072.
Intersection of A046100 and A013929.
Subsequence of A252849 and A375229.
Cf. A004709, A005117, A051903, A059956, A215267.

Select[Range[200], 2 <= Max[FactorInteger[#][[;; , 2]]] <= 3 &]

isok(k) = if(k == 1, 0, my(emax = vecmax(factor(k)[, 2])); emax > 1 & emax < 4);

from math import isqrt
from sympy import mobius, integer_nthroot
def A382967(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+sum(mobius(k)*(x//k**2-x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,4)[0]+1,isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Apr 11 2025

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (12*(15 - Pi^2)). - Vaclav Kotesovec, Apr 11 2025

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A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

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A382966 The number of non-unitary prime divisors of the n-th biquadratefree number that is not cubefree.

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A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

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A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).

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