cp's OEIS Frontend

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

%I A386802 #7 Aug 03 2025 16:09:52
%S A386802 27000,74088,189000,287496,297000,343000,351000,370440,459000,474552,
%T A386802 513000,621000,783000,814968,837000,963144,999000,1029000,1061208,
%U A386802 1107000,1157625,1161000,1259496,1269000,1323000,1331000,1407672,1431000,1437480,1481544,1593000,1647000
%N A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.
%C A386802 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.
%C A386802 Numbers k such that A295883(k) = 3.
%C A386802 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).
%H A386802 Amiram Eldar, <a href="/A386802/b386802.txt">Table of n, a(n) for n = 1..10000</a>
%H A386802 Ertan Elma and Greg Martin, <a href="https://doi.org/10.4153/S0008439524000584">Distribution of the number of prime factors with a given multiplicity</a>, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; <a href="https://arxiv.org/abs/2406.04574">arXiv preprint</a>, arXiv:2406.04574 [math.NT], 2024.
%H A386802 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386802 f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[2*10^6], s[#] == 3 &]
%o A386802 (PARI) isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 3;
%Y A386802 Subsequence of A176359.
%Y A386802 Cf. A295883.
%Y A386802 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).
%Y A386802 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).
%K A386802 nonn,easy
%O A386802 1,1
%A A386802 _Amiram Eldar_, Aug 03 2025