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A386806 Numbers that have exactly three exponents in their prime factorization that are equal to 4.

%I A386806 #6 Aug 03 2025 16:09:45
%S A386806 810000,3111696,5670000,8910000,10530000,13770000,15390000,15558480,
%T A386806 18630000,18974736,23490000,24010000,25110000,29970000,33210000,
%U A386806 34228656,34830000,37015056,38070000,39690000,40452048,42930000,47790000,49410000,52898832,54270000,57510000
%N A386806 Numbers that have exactly three exponents in their prime factorization that are equal to 4.
%C A386806 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^4 + 1/p^5) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 4.77477224068657540815...*10^(-7), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^5/(p-1)-1))^m (Elma and Martin, 2024).
%H A386806 Amiram Eldar, <a href="/A386806/b386806.txt">Table of n, a(n) for n = 1..10000</a>
%H A386806 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 A386806 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386806 seq[lim_] := Module[{s = {}, sqfs = Select[Range[Surd[lim, 4]], SquareFreeQ[#] && PrimeNu[#] == 3 &]}, Do[s = Join[s, sqf^4 * Select[Range[lim/sqf^4], CoprimeQ[#, sqf] && !MemberQ[FactorInteger[#][[;; , 2]], 4] &]], {sqf, sqfs}]; Union[s]]; seq[6*10^7]
%o A386806 (PARI) isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 3;
%Y A386806 Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), A386802 (k=3), this sequence (k=4), A386810 (k=5).
%Y A386806 Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), A386804 (m=1), A386805 (m=2), this sequence (m=3).
%K A386806 nonn,easy
%O A386806 1,1
%A A386806 _Amiram Eldar_, Aug 03 2025