A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268

Offset: 1

Author

Amiram Eldar, Aug 02 2025

Keywords

• nonn
• easy

Comments

First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.

Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.

Numbers k such that A369427(k) = 1.

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).

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.
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

A060687 is a subsequence.

Cf. A286228, A330596, A369427.

Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).

Programs

Mathematica

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]

PARI

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