cp's OEIS Frontend

A386800 Numbers that have exactly one exponent in their prime factorization that is equal to 3.

%I A386800 #8 Aug 13 2025 07:17:02
%S A386800 8,24,27,40,54,56,72,88,104,108,120,125,135,136,152,168,184,189,200,
%T A386800 232,248,250,264,270,280,296,297,312,328,343,344,351,360,375,376,378,
%U A386800 392,408,424,432,440,456,459,472,488,500,504,513,520,536,540,552,568,584
%N A386800 Numbers that have exactly one exponent in their prime factorization that is equal to 3.
%C A386800 First differs from its subsequence A381315 at n = 40: a(40) = 432 = 2^4 * 3^3 is not a term of A381315.
%C A386800 Numbers k such that A295883(k) = 1.
%C A386800 The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + 1/p^4) * Sum_{p prime} (p-1)/(p^4 - p + 1) = 0.092831691827595439609... (Elma and Martin, 2024).
%H A386800 Amiram Eldar, <a href="/A386800/b386800.txt">Table of n, a(n) for n = 1..10000</a>
%H A386800 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 A386800 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386800 f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]
%o A386800 (PARI) isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 1;
%Y A386800 A381315 is subsequence.
%Y A386800 Cf. A295883.
%Y A386800 Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), A386796 (k=2), this sequence (k=3), A386804 (k=4), A386808 (k=5).
%Y A386800 Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), this sequence (m=1), A386801 (m=2), A386802 (m=3).
%K A386800 nonn,easy
%O A386800 1,1
%A A386800 _Amiram Eldar_, Aug 03 2025