A386803 Numbers without an exponent 4 in their prime factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 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 A209061 at n = 246: a(246) = 256 = 2^8 is not a term of A209061.

First differs from its subsequences A115063 and A369939 at n = 62: a(62) = 64 = 2^6 is not a term of A115063.

The complement of this sequence is a subsequence of A336595.

These numbers were named semi-4-free integers by Suryanarayana (1971).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) = 0.95908865419555719109... (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]], 4] &]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 0;

Crossrefs

Subsequences: A115063, A209061, A369939.

Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), this sequence (k=4), A386807 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 4: this sequence (m=0), A386804 (m=1), A386805 (m=2), A386806 (m=3).

Sequence in context: A194897 A140823 A209061 * A388972 A390438 A115063

Adjacent sequences: A386800 A386801 A386802 * A386804 A386805 A386806

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved