A386807 Numbers without an exponent 5 in their prime factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset

1,2

Comments

First differs from its subsequence A166718 at n = 47: a(47) = 48 = 2^4 * 3 is not a term of A166718.

Differs from A373868 by having the terms 1, 1024, 32768, 59049, ..., and not having the terms 96, 160, 224, ... .

These numbers were named semi-5-free integers by Suryanarayana (1971).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.98136375107187963656... (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]], 5] &]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 0;

Crossrefs

A166718 is a subsequence.

Cf. A373868.

Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), A386803 (k=4), this sequence (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 5: this sequence (m=0), A386808 (m=1), A386809 (m=2), A386810 (m=3).

Sequence in context: A257333 A090110 A373868 * A166718 A132016 A032513

Adjacent sequences: A386804 A386805 A386806 * A386808 A386809 A386810

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved