A337050 Numbers without an exponent 2 in their prime factorization.

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, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87

Offset

1,2

Comments

Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).

Numbers k such that A003557(k) = k/A007947(k) is a powerful number (A001694).

The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).

A304364 is apparently a subsequence.

These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.

D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.

D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.

Formula

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

Example

6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.

Maple

q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
select(q, [$1..100])[];  # Alois P. Heinz, Aug 12 2020

Mathematica

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 2] &]

Prog

(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023

Crossrefs

Complement of A038109.

Cf. A003557, A007947, A057521, A304364, A330596.

A005117, A036537, A036966, A048109, A175496, A268335 and A336590 are subsequences.

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).

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).

Sequence in context: A028764 A366319 A326070 * A304364 A336590 A000452

Adjacent sequences: A337047 A337048 A337049 * A337051 A337052 A337053

Keyword

nonn,easy

Author

Amiram Eldar, Aug 12 2020

Status

approved