cp's OEIS Frontend

A386809 Numbers that have exactly two exponents in their prime factorization that are equal to 5.

%I A386809 #7 Aug 03 2025 16:05:25
%S A386809 7776,38880,54432,85536,100000,101088,132192,147744,178848,194400,
%T A386809 225504,241056,272160,287712,300000,318816,334368,365472,381024,
%U A386809 412128,427680,458784,474336,505440,520992,537824,552096,567648,598752,614304,645408,660960,692064,700000
%N A386809 Numbers that have exactly two exponents in their prime factorization that are equal to 5.
%C A386809 The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * ((Sum_{p prime} (p-1)/(p^6 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^6 - p + 1)^2)) / 2 = 4.86539910559896710587...*10^(-5) (Elma and Martin, 2024).
%H A386809 Amiram Eldar, <a href="/A386809/b386809.txt">Table of n, a(n) for n = 1..10000</a>
%H A386809 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 A386809 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%t A386809 f[p_, e_] := If[e == 5, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[700000], s[#] == 2 &]
%o A386809 (PARI) isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 2;
%Y A386809 Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), A386801 (k=3), A386805 (k=4), this sequence (k=5).
%Y A386809 Numbers that have exactly m exponents in their prime factorization that are equal to 5: A386807 (m=0), A386808 (m=1), this sequence (m=2), A386810 (m=3).
%K A386809 nonn,easy
%O A386809 1,1
%A A386809 _Amiram Eldar_, Aug 03 2025