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A386810 Numbers that have exactly three exponents in their prime factorization that are equal to 5.

%I A386810 #11 Aug 03 2025 16:05:43
%S A386810 24300000,130691232,170100000,267300000,315900000,413100000,461700000,
%T A386810 558900000,653456160,704700000,753300000,899100000,996300000,
%U A386810 1044900000,1142100000,1190700000,1252332576,1287900000,1433700000,1437603552,1482300000,1628100000,1680700000
%N A386810 Numbers that have exactly three exponents in their prime factorization that are equal to 5.
%C A386810 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 1.38560245036673575581*10^(-8), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^6/(p-1)-1))^m (Elma and Martin, 2024).
%H A386810 Amiram Eldar, <a href="/A386810/b386810.txt">Table of n, a(n) for n = 1..10000</a>
%H A386810 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 A386810 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%p A386810 M:= 10^10: # for terms <= M
%p A386810 B:= select(t -> ifactors(t)[2][..,2]=[1,1,1],[$1..floor(M^(1/5))]):
%p A386810 R:= NULL:
%p A386810 for i from 1 to nops(B) do
%p A386810   Q:= select(t -> igcd(t,B[i]) = 1 and not member(5, ifactors(t)[2][..,2]), [$1 .. M/B[i]^5]);
%p A386810   R:= R, op(B[i]^5 * Q);
%p A386810 od:
%p A386810 sort([R]); # _Robert Israel_, Aug 03 2025
%t A386810 seq[lim_] := Module[{s = {}, sqfs = Select[Range[Surd[lim, 5]], SquareFreeQ[#] && PrimeNu[#] == 3 &]}, Do[s = Join[s, sqf^5 * Select[Range[lim/sqf^5], CoprimeQ[#, sqf] && !MemberQ[FactorInteger[#][[;; , 2]], 5] &]], {sqf, sqfs}]; Union[s]]; seq[2*10^9]
%o A386810 (PARI) isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 3;
%Y A386810 Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), A386802 (k=3), A386806 (k=4), this sequence (k=5).
%Y A386810 Numbers that have exactly m exponents in their prime factorization that are equal to 5: A386807 (m=0), A386808 (m=1), A386809 (m=2), this sequence (m=3).
%K A386810 nonn,easy
%O A386810 1,1
%A A386810 _Amiram Eldar_, Aug 03 2025