cp's OEIS Frontend

A377590 Numbers k neither squarefree nor prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.

%I A377590 #13 Jun 21 2025 19:59:44
%S A377590 12,24,45,48,63,75,96,135,175,189,192,225,245,275,325,384,405,425,475,
%T A377590 539,567,575,605,637,675,768,833,847,875,931,1127,1183,1215,1225,1375,
%U A377590 1421,1519,1536,1573,1625,1701,1715,1813,1859,1925,2009,2023,2025,2057,2107
%N A377590 Numbers k neither squarefree nor prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.
%C A377590 This sequence contains numbers k in A126706 for which A376846(k) = 0; A376846(k) = 0 for prime powers k or squarefree numbers k (i.e., k in A303554).
%C A377590 It is sufficient to determine floor(log k / log p) <= Omega(k) for p = lpf(k) = A020639(k).
%C A377590 Sequence contains numbers k of the form 2^j*3, j > 1, i.e., A007283 \ {3, 6} is a proper subset of this sequence, since 2^(j+1) < 2^j*3 and j+1 = Omega(2^j*3).
%C A377590 The numbers k that remain in the sequence ({a(n)} \ A007283) are odd, that is, in A360769. For k = 2^j*p, prime p > 3, we have j+floor(log_2 p) > j+1, since log_2 p > 2, therefore we see m = 2^(j+floor(log_2 p)) < 2^j*p, with Omega(m) > Omega(k).
%H A377590 Michael De Vlieger, <a href="/A377590/b377590.txt">Table of n, a(n) for n = 1..10000</a>
%H A377590 Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.18645.64480">Numbers k for which floor(log k / log lpf(k)) <= bigomega(k)</a>, 2024.
%e A377590 12 is in the sequence since 2^3 < 12, and Omega(2^3) = Omega(12) = 3.
%e A377590 20 is not in the sequence since 2^4 < 20 and Omega(2^4) = 4, but Omega(20) = 3.
%e A377590 45 is in the sequence since 3^3 < 45, and Omega(3^3) = Omega(45) = 3.
%e A377590 375 = 3*5^3 is not in the sequence since 3^5 < 375 and Omega(3^5) = 5, but Omega(345) = 4.
%t A377590 Select[Select[Range[4000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{n, k}, NoneTrue[FactorInteger[n][[All, 1]], Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]
%Y A377590 Cf. A001222, A007283, A007947, A020639, A126706, A303554, A360769.
%K A377590 nonn,easy
%O A377590 1,1
%A A377590 _Michael De Vlieger_, Nov 02 2024