A336592 -id:A336592 - cp's OEIS frontend

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

Amiram Eldar, Aug 02 2025

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).

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.

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).

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

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

A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 128, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 384, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that at least one of the exponents in their prime factorization is of the form 4*m + 3.
The asymptotic density of this sequence is 1 - zeta(4)/zeta(3) = 0.0996073223... (Cohen, 1963).
The number of divisors of all the terms is divisible by 4.

			8 is a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Complement of A336592.
Complement of A336594 within A252849.
Cf. A002117, A008835, A013662, A036966.
A176297 is a subsequence.

Select[Range[540], Max[Mod[FactorInteger[#][[;; , 2]], 4]] == 3 &]

cp's OEIS Frontend

A386799 Numbers without an exponent 3 in their prime factorization.

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