cp's OEIS Frontend

A337050 Numbers without an exponent 2 in their prime factorization.

%I A337050 #28 Aug 03 2025 10:20:59
%S A337050 1,2,3,5,6,7,8,10,11,13,14,15,16,17,19,21,22,23,24,26,27,29,30,31,32,
%T A337050 33,34,35,37,38,39,40,41,42,43,46,47,48,51,53,54,55,56,57,58,59,61,62,
%U A337050 64,65,66,67,69,70,71,73,74,77,78,79,80,81,82,83,85,86,87
%N A337050 Numbers without an exponent 2 in their prime factorization.
%C A337050 Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).
%C A337050 Numbers k such that A003557(k) = k/A007947(k) is a powerful number (A001694).
%C A337050 The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
%C A337050 A304364 is apparently a subsequence.
%C A337050 These numbers were named semi-2-free integers by Suryanarayana (1971). - _Amiram Eldar_, Dec 29 2020
%H A337050 Amiram Eldar, <a href="/A337050/b337050.txt">Table of n, a(n) for n = 1..10000</a>
%H A337050 Ertan Elma and Greg Martin, <a href="https://doi.org/10.4153/S0008439524000584">Distribution of the number of prime factors with a given multiplicity</a>, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; <a href="https://arxiv.org/abs/2406.04574">arXiv preprint</a>, arXiv:2406.04574 [math.NT], 2024.
%H A337050 D. Suryanarayana, <a href="https://eudml.org/doc/141006">Semi-k-free integers</a>, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
%H A337050 D. Suryanarayana and R. Sitaramachandra Rao, <a href="https://doi.org/10.1090/S0002-9939-1973-0311599-1">Distribution of semi-k-free integers</a>, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
%F A337050 Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - _Amiram Eldar_, Oct 21 2023
%e A337050 6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
%e A337050 9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.
%p A337050 q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
%p A337050 select(q, [$1..100])[];  # _Alois P. Heinz_, Aug 12 2020
%t A337050 Select[Range[100], !MemberQ[FactorInteger[#][[;;, 2]], 2] &]
%o A337050 (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ _Amiram Eldar_, Oct 21 2023
%Y A337050 Complement of A038109.
%Y A337050 Cf. A003557, A007947, A057521, A304364, A330596.
%Y A337050 A005117, A036537, A036966, A048109, A175496, A268335 and A336590 are subsequences.
%Y A337050 Numbers without an exponent k in their prime factorization: A001694 (k=1), this sequence (k=2), A386799 (k=3), A386803 (k=4), A386807 (k=5).
%Y A337050 Numbers that have exactly m exponents in their prime factorization that are equal to 2: this sequence (m=0), A386796 (m=1), A386797 (m=2), A386798 (m=3).
%K A337050 nonn,easy
%O A337050 1,2
%A A337050 _Amiram Eldar_, Aug 12 2020