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A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.

%I A386801 #7 Aug 03 2025 16:09:55
%S A386801 216,1000,1080,1512,2376,2744,2808,3000,3375,3672,4104,4968,5400,6264,
%T A386801 6696,6750,7000,7560,7992,8232,8856,9000,9261,9288,10152,10584,10648,
%U A386801 11000,11448,11880,12744,13000,13176,13500,13720,14040,14472,15336,15768,16632,17000
%N A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.
%C A386801 Subsequence of A109399 and first differs from it at n = 64: A109399(64) = 27000 = 2^3 * 3^3 * 5^3 is not a term of this sequence.
%C A386801 Numbers k such that A295883(k) = 2.
%C A386801 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * ((Sum_{p prime} (p-1)/(p^4 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^4 - p + 1)^2)) / 2 = 0.0024403883082851652103... (Elma and Martin, 2024).
%H A386801 Amiram Eldar, <a href="/A386801/b386801.txt">Table of n, a(n) for n = 1..10000</a>
%H A386801 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 A386801 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386801 f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[17000], s[#] == 2 &]
%o A386801 (PARI) isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 2;
%Y A386801 Subsequence of A109399.
%Y A386801 Cf. A295883.
%Y A386801 Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), this sequence (k=3), A386805 (k=4), A386809 (k=5).
%Y A386801 Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), this sequence (m=2), A386802 (m=3).
%K A386801 nonn,easy
%O A386801 1,1
%A A386801 _Amiram Eldar_, Aug 03 2025