A362148 -id:A362148 - cp's OEIS frontend

A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384

Offset: 1

Michael De Vlieger, Dec 06 2024

Products m = j*k such that omega(k) = omega(m) > omega(j), where rad(j) | k but j does not divide k, with rad = A007947 and omega = A001221.
Proper subset of A126706.
This sequence is distinct from A362148, since this sequence also contains 216, 432, etc.

			Prime decomposition of select a(n) = m, showing m = j*k:
a(1) = 24 = 2^3 * 3 = 4 * 6.
a(2) = 40 = 2^3 * 5 = 4 * 10.
a(3) = 48 = 2^4 * 3 = 8 * 6.
a(4) = 54 = 2 * 3^3 = 9 * 6.
a(5) = 56 = 2^3 * 7 = 4 * 14.
a(6) = 72 = 2^3 * 3^2 = 4 * 18.
a(9) = 96 = 2^5 * 3 = 8 * 12 = 16 * 6.
a(130) = 864 = 2^5 * 3^2 = 8 * 108 = 9 * 96 = 16 * 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A024619, A046099, A126706, A362148, A372695.

Select[Select[Range[2^10], AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &], Not@*PrimePowerQ]

{a(n)} = { k : omega(k) > 1, there exists p | k such that p^3 | k }.
Intersection of A046099 and A024619.
Union of A362148 and A372695.

A362147 Numbers that are not cubefull.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Bernard Schott, Apr 09 2023

nonn

Integers m for which there is a prime p that divides m, but p^3 does not divide m.

Complement of A036966.

			2|24 and 2^3|24, but 3|24 and 3^3 does not divide 24, so 24 is a term.

Cf. A004709 (cubefree), A046099 (not cubefree), A036966 (cubefull), A362148 (non-cubefree that are not cubefull).

Select[Range[2, 100], Min[FactorInteger[#][[;; , 2]]] < 3 &] (* Amiram Eldar, Apr 09 2023 *)

isok(k) = (k!=1) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 12 2023

from math import gcd
from sympy import integer_nthroot, factorint
def A362147(n):
    def f(x):
        c = n
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Nov 22 2024

cp's OEIS Frontend

A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

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