A166718 -id:A166718 - cp's OEIS frontend

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

Amiram Eldar, Aug 03 2025

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).

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.

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).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 5] &]

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

A166719 Numbers with at most 5 prime factors (counted with multiplicity).

Original entry on oeis.org

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, 32, 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, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Michael B. Porter, Oct 20 2009

Complement of A046305, A001222(a(n))<=5

			50 = 2*5*5 is in the sequence since it has 3 prime factors and 3 <= 5
64 = 2*2*2*2*2*2 is not in the sequence since it has 6 prime factors

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A046305, A001222

For numbers with at most n prime factors: n=1: A000040, n=2: A037143, n=3: A037144, n=4: A166718.

Select[Range[100],PrimeOmega[#]<6&] (* Harvey P. Dale, Jul 13 2011 *)

PARI
```
isA166719(n) = (bigomega(n) <= 5)
```

cp's OEIS Frontend

A386807 Numbers without an exponent 5 in their prime factorization.

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