A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.

27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000

Offset: 1

Author

Amiram Eldar, Aug 03 2025

Keywords

• nonn
• easy

Comments

Subsequence of A176359 and first differs from it at n = 173: A176359(173) = 9261000 = 2^3 * 3^3 * 5^3 * 7^3 is not a term of this sequence.

Numbers k such that A295883(k) = 3.

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.000018940548516752487509..., where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^4/(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

Subsequence of A176359.

Cf. A295883.

Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), this sequence (k=3), A386806 (k=4), A386810 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), A386801 (m=2), this sequence (m=3).

Programs

Mathematica

f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[2*10^6], s[#] == 3 &]

PARI

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