login
Numbers that are neither cubefree nor cubefull.
3

%I #61 Dec 06 2024 13:17:55

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

%T 168,176,184,189,192,200,208,224,232,240,248,250,264,270,272,280,288,

%U 296,297,304,312,320,324,328,336,344,351,352,360,368,375,376,378,384,392,400

%N Numbers that are neither cubefree nor cubefull.

%C In fact, every cubefull number > 1 is noncubefree, but the converse is false.

%C This sequence = A046099 \ A036966 and lists these counterexamples.

%C Numbers k such that for some primes p and q, k is divisible by p^3*q but not by q^3. - _Robert Israel_, Apr 28 2023

%C The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - _Charles R Greathouse IV_, Apr 28 2023

%C From _Amiram Eldar_, Sep 17 2023: (Start)

%C Numbers k such that A360539(k) > 1 and A360540(k) > 1.

%C Equivalently, numbers that have in their prime factorization at least one exponent that is smaller than 3 and at least one exponent that is larger than 2. (End)

%H Michael De Vlieger, <a href="/A362148/b362148.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals A362147 \ A004709.

%F Sum_{n>=1} 1/a(n) = 1 + zeta(s) - zeta(s)/zeta(3*s) - Product_{p prime}(1 + 1/(p^(2*s)*(p^s-1))), s > 1. - _Amiram Eldar_, Sep 17 2023

%e 24 = 2^3 * 3 is noncubefree as it is divisible by the cube 2^3, but it is not cubefull because 3 divides 24 but 3^3 does not divide 24, hence 24 is a term.

%e 648 = 2^4 * 3^3 is noncubefree as it is divisible by the cube 3^3, but it is also cubefull because primes 2 and 3 divide 648, and 2^3 and 3^3 divide also 648, so 648 is not a term.

%p filter:= proc(n) local F;

%p F:= ifactors(n)[2][..,2];

%p min(F) < 3 and max(F) >= 3

%p end proc:

%p select(filter, [$1..400]); # _Robert Israel_, Apr 28 2023

%t Select[Range[500], Min[(e = FactorInteger[#][[;; , 2]])] < 3 && Max[e] > 2 &] (* _Amiram Eldar_, Apr 09 2023 *)

%o (PARI) isok(k) = (k>1) && (vecmax(factor(k)[, 2])>2) && (vecmin(factor(k)[, 2])<=2); \\ _Michel Marcus_, Apr 19 2023

%Y Intersection of A046099 (not cubefree) and A362147 (not cubefull)

%Y Cf. A004709 (cubefree), A036966 (cubefull), A360539, A360540.

%K nonn,easy

%O 1,1

%A _Bernard Schott_, Apr 09 2023