A386799 Numbers without an exponent 3 in their prime factorization.

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, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset

1,2

Comments

First differs from its subsequence A336592 at n = 116: a(116) = 128 = 2^7 is not a term of A336592.

Numbers k such that A295883(k) = 0.

These numbers were named semi-3-free integers by Suryanarayana (1971).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.90470892696874750603... (Suryanarayana, 1971).

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.

Index entries for sequences computed from indices in prime factorization.

Mathematica

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 3] &]

Prog

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

Crossrefs

Complement of A176297.

A336592 is a subsequence.

Cf. A295883.

Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), this sequence (k=3), A386803 (k=4), A386807 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 3: this sequence (m=0), A386800 (m=1), A386801 (m=2), A386802 (m=3).

Sequence in context: A007412 A270420 A336592 * A390437 A392632 A335275

Adjacent sequences: A386796 A386797 A386798 * A386800 A386801 A386802

Keyword

nonn,easy

Author

Amiram Eldar, Aug 02 2025

Status

approved