This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375055 #23 Oct 25 2024 09:30:09
%S A375055 60,84,90,120,126,132,140,150,156,168,180,198,204,220,228,234,240,252,
%T A375055 260,264,270,276,280,294,300,306,308,312,315,336,340,342,348,350,360,
%U A375055 364,372,378,380,396,408,414,420,440,444,450,456,460,468,476,480,490,492,495
%N A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.
%C A375055 Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.
%C A375055 A178212 is a proper subset.
%C A375055 This sequence is distinct from A123712 since 420 is here.
%C A375055 This sequence is distinct from A182855 since 360 is here.
%H A375055 Michael De Vlieger, <a href="/A375055/b375055.txt">Table of n, a(n) for n = 1..10000</a>
%F A375055 {a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.
%e A375055 a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
%e A375055 72 is not in the sequence because it is the product of 2 distinct prime factors.
%e A375055 a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
%e A375055 a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
%e A375055 a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
%e A375055 210 is not in the sequence because it is squarefree.
%e A375055 a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
%e A375055 a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
%e A375055 .
%e A375055 Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
%e A375055 n a(n) pair of factors with quality Q.
%e A375055 -------------------------------------------------------------------
%e A375055 1 60 6 X 10;
%e A375055 2 84 6 X 14;
%e A375055 3 90 6 X 15;
%e A375055 4 120 6 X 20, 10 X 12;
%e A375055 5 126 6 X 21;
%e A375055 6 132 6 X 22;
%e A375055 7 140 10 X 14;
%e A375055 8 150 10 X 15;
%e A375055 17 240 6 X 40, 10 X 24, 12 X 20;
%e A375055 51 480 6 X 80, 10 X 48, 12 X 40, 20 X 24;
%e A375055 117 840 6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.
%t A375055 Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]
%Y A375055 Cf. A001221, A001222, A013929, A024619, A126706, A178212.
%K A375055 nonn,easy
%O A375055 1,1
%A A375055 _Michael De Vlieger_, Oct 22 2024