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A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

%I A386796 #10 Aug 03 2025 10:20:54
%S A386796 4,9,12,18,20,25,28,44,45,49,50,52,60,63,68,72,75,76,84,90,92,98,99,
%T A386796 108,116,117,121,124,126,132,140,144,147,148,150,153,156,164,169,171,
%U A386796 172,175,188,198,200,204,207,212,220,228,234,236,242,244,245,260,261,268
%N A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.
%C A386796 First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.
%C A386796 Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.
%C A386796 Numbers k such that A369427(k) = 1.
%C A386796 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).
%H A386796 Amiram Eldar, <a href="/A386796/b386796.txt">Table of n, a(n) for n = 1..10000</a>
%H A386796 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 A386796 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386796 f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]
%o A386796 (PARI) isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 1;
%Y A386796 A060687 is a subsequence.
%Y A386796 Cf. A286228, A330596, A369427.
%Y A386796 Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).
%Y A386796 Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).
%K A386796 nonn,easy
%O A386796 1,1
%A A386796 _Amiram Eldar_, Aug 02 2025