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

%I A386798 #7 Aug 03 2025 10:20:46
%S A386798 900,1764,4356,4900,6084,6300,8820,9900,10404,11025,11700,12100,12996,
%T A386798 14700,15300,16900,17100,19044,19404,20700,21780,22050,22932,23716,
%U A386798 26100,27225,27900,28900,29988,30276,30420,30492,33124,33300,33516,34596,36100,36300,36900,38025,38700
%N A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.
%C A386798 Numbers k such that A369427(k) = 2.
%C A386798 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).
%H A386798 Amiram Eldar, <a href="/A386798/b386798.txt">Table of n, a(n) for n = 1..10000</a>
%H A386798 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 A386798 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386798 f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[40000], s[#] == 3 &]
%o A386798 (PARI) isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 3;
%Y A386798 Cf. A330596, A369427.
%Y A386798 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).
%Y A386798 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).
%K A386798 nonn,easy
%O A386798 1,1
%A A386798 _Amiram Eldar_, Aug 02 2025