cp's OEIS Frontend

A386805 Numbers that have exactly two exponents in their prime factorization that are equal to 4.

%I A386805 #6 Aug 03 2025 16:09:42
%S A386805 1296,6480,9072,10000,14256,16848,22032,24624,29808,30000,32400,37584,
%T A386805 38416,40176,45360,47952,50625,53136,55728,60912,63504,68688,70000,
%U A386805 71280,76464,79056,84240,86832,90000,92016,94608,99792,101250,102384,107568,110000,110160,115248
%N A386805 Numbers that have exactly two exponents in their prime factorization that are equal to 4.
%C A386805 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^4 + 1/p^5) * ((Sum_{p prime} (p-1)/(p^5 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^5 - p + 1)^2)) / 2 = 0.00032582100547959312658... (Elma and Martin, 2024).
%H A386805 Amiram Eldar, <a href="/A386805/b386805.txt">Table of n, a(n) for n = 1..10000</a>
%H A386805 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 A386805 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386805 f[p_, e_] := If[e == 4, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[120000], s[#] == 2 &]
%o A386805 (PARI) isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 2;
%Y A386805 Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), A386801 (k=3), this sequence (k=4), A386809 (k=5).
%Y A386805 Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), A386804 (m=1), this sequence (m=2), A386806 (m=3).
%K A386805 nonn,easy
%O A386805 1,1
%A A386805 _Amiram Eldar_, Aug 03 2025