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

Author

Amiram Eldar, Aug 02 2025

Keywords

• nonn
• easy

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.

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).

Programs

Mathematica

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

PARI

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