A335438 - OEIS

%I #13 Dec 26 2024 10:21:17

%S 1,1,1,2,1,1,2,1,1,3,2,1,1,3,2,2,1,4,1,1,3,1,4,2,1,1,5,2,1,3,1,5,3,2,

%T 1,1,4,6,1,2,1,2,1,3,6,1,4,1,1,2,1,7,1,1,5,4,3,2,2,7,1,1,1,2,1,5,8,3,

%U 1,4,1,1,1,3,8,2,1,1,6,1,3,2,1,1,2,9,5,1,1,2,1,3

%N Number of partitions of k_n into two distinct parts (s,t) such that k_n | s*t, where k_n = A335437(n).

%C a(n) >= 1.

%H Robert Israel, <a href="/A335438/b335438.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = floor((A000188(A335437(n))-1)/2). - _Robert Israel_, Dec 23 2024

%e a(2) = 1; A335437(2) = 16 has exactly one partition into two distinct parts (12,4), such that 16 | 12*4 = 48. Therefore, a(2) = 1.

%p f:= proc(n) local F,beta,t;

%p      F:= ifactors(n)[2];

%p      beta:= mul(t[1]^floor(t[2]/2),t=F);

%p      if beta <= 2 then NULL else floor((beta-1)/2) fi

%p end proc:

%p map(f, [$1..500]); # _Robert Israel_, Dec 23 2024

%t Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}], {}], {n, 400}] // Flatten

%Y Cf. A000188, A013929, A335234, A335437.

%K nonn

%O 1,4

%A _Wesley Ivan Hurt_, Jun 10 2020