A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.

900, 1764, 4356, 4900, 6084, 6300, 8820, 9900, 10404, 11025, 11700, 12100, 12996, 14700, 15300, 16900, 17100, 19044, 19404, 20700, 21780, 22050, 22932, 23716, 26100, 27225, 27900, 28900, 29988, 30276, 30420, 30492, 33124, 33300, 33516, 34596, 36100, 36300, 36900, 38025, 38700

Offset: 1

Author

Amiram Eldar, Aug 02 2025

Keywords

• nonn
• easy

Comments

Numbers k such that A369427(k) = 2.

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.0011175284878980531468... (the product is A330596), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^3/(p-1)-1))^m (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

Cf. A330596, A369427.

Numbers that have exactly three exponents in their prime factorization that are equal to k: this sequence (k=2), A386802 (k=3), A386806 (k=4), A386810 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), A386796 (m=1), A386797 (m=2), this sequence (m=3).

Programs

Mathematica

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[40000], s[#] == 3 &]

PARI

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