A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

24, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272, 276, 280, 288, 294

Offset

1,1

Comments

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.

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

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:

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.

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

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

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Chart of (d, a(n)/d) 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.

Formula

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.

Union of A375055, A376936, and A378767.

Example

a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).
a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).
a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).
a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).
a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.
a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)
a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).
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.

Mathematica

nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
Complement[
  Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p] ] ][[-1, 1]] ]^2 ]

Crossrefs

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

Sequence in context: A210456 A095158 A271422 * A378767 A362148 A062374

Adjacent sequences: A379333 A379334 A379335 * A379337 A379338 A379339

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 24 2024

Status

approved