A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116

Offset: 1

Author

Wesley Ivan Hurt, Jan 16 2022

Keywords

• nonn

Comments

From Bernard Schott, Jan 22 2022: (Start)

A299174 is a subsequence because, if k = 2*u, we have s=t=u, s<=t, and u | u*k.

A082663 is another subsequence because, if k = p*q with p < q < 2p, then with s = k-p^2 = p*(q-p) and t = p^2, we have s <= t and p^2 | p*(q-p) * (pq).

It seems that A090196 is the subsequence of odd terms. (End)

gcd(s, t) > 1 where s and t and k > 2 are as in name. - David A. Corneth, Jan 22 2022

Numbers k such that k^2 has at least one divisor d with k/2 <= d < k. - Robert Israel, Jan 08 2025

Examples

			15 is in the sequence since 15 = 6+9 where 9 | 6*15 = 90.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A338021, A350804 (exactly one).

Subsequences: A082663, A299174.

Cf. A090196.

Programs

Maple

filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))>=1 end proc:
select(filter, [$1..300]); # Robert Israel, Jan 08 2025

PARI

f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
isok(k) = f(k) >= 1; \\ Michel Marcus, Jan 17 2022