A386797 Numbers that have exactly two exponents in their prime factorization that are equal to 2.

36, 100, 180, 196, 225, 252, 300, 396, 441, 450, 468, 484, 588, 612, 676, 684, 700, 828, 882, 980, 1044, 1089, 1100, 1116, 1156, 1225, 1260, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1692, 1700, 1800, 1900, 1908, 1980, 2028, 2100, 2116, 2124, 2156, 2178, 2196

Offset: 1

Author

Amiram Eldar, Aug 02 2025

Keywords

• nonn
• easy

Comments

First differs from its subsequence A375144 at n = 38: a(38) = 1800 = 2^3 * 3^2 * 5^2 is not a term of A375144.

Numbers k such that A369427(k) = 2.

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))^2 - Sum_{p prime} ((p-1)^2/(p^3 - p + 1)^2)) / 2 = 0.023701044250873975412... (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

A375144 is a subsequence.

Cf. A330596, A369427.

Numbers that have exactly two exponents in their prime factorization that are equal to k: this sequence (k=2), A386801 (k=3), A386805 (k=4), A386809 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), A386796 (m=1), this sequence (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[2200], s[#] == 2 &]

PARI

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