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

%I A350803 #31 Jan 09 2025 02:22:05
%S A350803 2,4,6,8,10,12,14,15,16,18,20,22,24,26,28,30,32,34,35,36,38,40,42,44,
%T A350803 45,46,48,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,77,78,80,82,
%U A350803 84,86,88,90,91,92,94,96,98,99,100,102,104,105,106,108,110,112,114,116
%N A350803 Numbers k with at least one partition into two parts (s,t), s<=t such that t | s*k.
%C A350803 From _Bernard Schott_, Jan 22 2022: (Start)
%C A350803 A299174 is a subsequence because, if k = 2*u, we have s=t=u, s<=t, and u | u*k.
%C A350803 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).
%C A350803 It seems that A090196 is the subsequence of odd terms. (End)
%C A350803 gcd(s, t) > 1 where s and t and k > 2 are as in name. - _David A. Corneth_, Jan 22 2022
%C A350803 Numbers k such that k^2 has at least one divisor d with k/2 <= d < k. - _Robert Israel_, Jan 08 2025
%H A350803 Robert Israel, <a href="/A350803/b350803.txt">Table of n, a(n) for n = 1..10000</a>
%H A350803 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%e A350803 15 is in the sequence since 15 = 6+9 where 9 | 6*15 = 90.
%p A350803 filter:= proc(n) nops(select(t -> t >= n/2 and t < n, numtheory:-divisors(n^2)))>=1 end proc:
%p A350803 select(filter, [$1..300]); # _Robert Israel_, Jan 08 2025
%o A350803 (PARI) f(n) = sum(s=1, n\2, !((s*n)%(n-s))); \\ A338021
%o A350803 isok(k) = f(k) >= 1; \\ _Michel Marcus_, Jan 17 2022
%Y A350803 Cf. A338021, A350804 (exactly one).
%Y A350803 Subsequences: A082663, A299174.
%Y A350803 Cf. A090196.
%K A350803 nonn
%O A350803 1,1
%A A350803 _Wesley Ivan Hurt_, Jan 16 2022