A378767 - OEIS

%I #5 Dec 08 2024 17:28:17

%S 24,40,48,54,56,72,80,88,96,104,108,112,120,135,136,144,152,160,162,

%T 168,176,184,189,192,200,208,216,224,232,240,248,250,264,270,272,280,

%U 288,296,297,304,312,320,324,328,336,344,351,352,360,368,375,376,378,384

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

%C 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.

%C Proper subset of A126706.

%C This sequence is distinct from A362148, since this sequence also contains 216, 432, etc.

%H Michael De Vlieger, <a href="/A378767/b378767.txt">Table of n, a(n) for n = 1..10000</a>

%F {a(n)} = { k : omega(k) > 1, there exists p | k such that p^3 | k }.

%F Intersection of A046099 and A024619.

%F Union of A362148 and A372695.

%e Prime decomposition of select a(n) = m, showing m = j*k:

%e a(1) = 24 = 2^3 * 3 = 4 * 6.

%e a(2) = 40 = 2^3 * 5 = 4 * 10.

%e a(3) = 48 = 2^4 * 3 = 8 * 6.

%e a(4) = 54 = 2 * 3^3 = 9 * 6.

%e a(5) = 56 = 2^3 * 7 = 4 * 14.

%e a(6) = 72 = 2^3 * 3^2 = 4 * 18.

%e a(9) = 96 = 2^5 * 3 = 8 * 12 = 16 * 6.

%e a(130) = 864 = 2^5 * 3^2 = 8 * 108 = 9 * 96 = 16 * 54, etc.

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

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

%K nonn,easy,new

%O 1,1

%A _Michael De Vlieger_, Dec 06 2024