A390437 - OEIS

%I #10 Nov 06 2025 00:14:57

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,28,

%T 29,30,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,

%U 53,55,57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76

%N Numbers whose prime factorization exponents are not divisible by 3.

%C Subsequence of A386799 and first differs from it at n = 58: A386799(58) = 64 = 2^6 is not a term in this sequence.

%C First differs from A336592 at n = 58: A336592(58) = 64 = 2^6 is not a term in this sequence. Also, a(115) = 128 = 2^7 is the least term that is not a term in A336592.

%C A182120 and A366762 are subsequences. Each term in this sequence has a unique representation as the product of two coprime numbers, one in A182120 and the other in A366762.

%C Numbers whose unitary divisors (A077610) that are larger than 1 are all noncubes (A007412).

%C The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p*(1+p+p^2))) = 0.95692470821076622881... .

%H Amiram Eldar, <a href="/A390437/b390437.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 1/p^(3*s)) for s > 1.

%t q[n_] := AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[100], q]

%o (PARI) isok(k) = vecsum(apply(x -> if(x % 3, 0, 1), factor(k)[, 2])) == 0;

%Y Subsequence of A386799.

%Y Subsequences: A001651, A005117, A004709, A182120, A366762.

%Y Cf. A007412, A077610.

%Y Numbers without exponents that are divisible by m: A268335 (m = 2), this sequence (m = 3), A390438 (m = 4).

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Nov 05 2025