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A362148
Numbers that are neither cubefree nor cubefull.
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, 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, 392, 400
OFFSET
1,1
COMMENTS
In fact, every cubefull number > 1 is noncubefree, but the converse is false.
This sequence = A046099 \ A036966 and lists these counterexamples.
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
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Charles R Greathouse IV, Apr 28 2023
From Amiram Eldar, Sep 17 2023: (Start)
Numbers k such that A360539(k) > 1 and A360540(k) > 1.
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)
FORMULA
Equals A362147 \ A004709.
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
EXAMPLE
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.
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.
MAPLE
filter:= proc(n) local F;
F:= ifactors(n)[2][.., 2];
min(F) < 3 and max(F) >= 3
end proc:
select(filter, [$1..400]); # Robert Israel, Apr 28 2023
MATHEMATICA
Select[Range[500], Min[(e = FactorInteger[#][[;; , 2]])] < 3 && Max[e] > 2 &] (* Amiram Eldar, Apr 09 2023 *)
PROG
(PARI) isok(k) = (k>1) && (vecmax(factor(k)[, 2])>2) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 19 2023
CROSSREFS
Intersection of A046099 (not cubefree) and A362147 (not cubefull)
Cf. A004709 (cubefree), A036966 (cubefull), A360539, A360540.
Sequence in context: A210456 A095158 A271422 * A062374 A272593 A048104
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Apr 09 2023
STATUS
approved