A335438 Number of partitions of k_n into two distinct parts (s,t) such that k_n | s*t, where k_n = A335437(n).

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

Offset: 1

Author

Wesley Ivan Hurt, Jun 10 2020

Keywords

• nonn

Comments

a(n) >= 1.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000188, A013929, A335234, A335437.

Programs

Maple

f:= proc(n) local F,beta,t;
     F:= ifactors(n)[2];
     beta:= mul(t[1]^floor(t[2]/2),t=F);
     if beta <= 2 then NULL else floor((beta-1)/2) fi
end proc:
map(f, [$1..500]); # Robert Israel, Dec 23 2024

Mathematica

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

Formula

a(n) = floor((A000188(A335437(n))-1)/2). - Robert Israel, Dec 23 2024