A379336 - OEIS

%I #19 Dec 26 2024 13:27:55

%S 24,40,48,54,56,60,72,80,84,88,90,96,104,108,112,120,126,132,135,136,

%T 140,144,150,152,156,160,162,168,176,180,184,189,192,198,200,204,208,

%U 216,220,224,228,232,234,240,248,250,252,260,264,270,272,276,280,288,294

%N Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

%C Both divisors d and d/k are composite, since primes p either divide or are coprime to another number, and all numbers smaller than p are coprime to p.

%C Proper subset of A126706; contains A378769, which in turn contains A378984.

%C Consider composite k, m, k != m. Define a "neutral" relation to be such that 1 < gcd(k,m) and not equal to either k or m. Then neither k nor m divides the other, and k and m are not coprime. If k is neutral to m, then m is neutral to k, since order does not matter. Then either the squarefree kernel of one divides the other or it does not. Thus, there are 3 kinds of neutral relation:

%C Type A: Though gcd(k,m) > 1, k has a factor P that does not divide m, and m has a factor Q that does not divide k.

%C Type B: rad(k) = rad(m), yet neither k divides m nor m divides k, where rad = A007947 is the squarefree kernel.

%C Type C: Squarefree kernel of one number divides the other, while the other has a factor that does not divide the former.

%C A378769, subset of this sequence, contains numbers k that have all 3 types of neutral relation between at least 1 divisor pair (d, k/d) for each.

%H Michael De Vlieger, <a href="/A379336/b379336.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A379336/a379336.png">Chart of (d, a(n)/d)</a> for a(n) = 1..144, showing only the smallest d for each type of neutral relation, where type A is shown in gray, type B in black, and type C in either blue or gold.

%F This sequence is A376271 \ A085986 = {k : bigomega(k) > omega(k) > 1, bigomega(k) > 3} \ { k^2 : bigomega(k) = omega(k) = 2 }, where bigomega = A001222 and omega = A001221.

%F Union of A375055, A376936, and A378767.

%e a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).

%e a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).

%e a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).

%e a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).

%e a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.

%e a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)

%e a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).

%e a(10475) = 32400 = 2^4 * 3^4 * 5^2 = 8*4050 (type C) = 48*675 (type A) = 120*270 (type B) = A378984(1) = A378769(14), etc.

%t nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;

%t Complement[

%t   Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],

%t   Union[Reap[

%t     While[p <= mm, q = NextPrime[p];

%t       While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];

%t         p = NextPrime[p] ] ][[-1, 1]] ]^2 ]

%Y Cf. A045763, A085986, A126706, A375055, A376271, A376936, A378767, A378769, A378984.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Dec 24 2024