A335234 Number of partitions of k_n into two parts (s,t) such that k_n | s*t, where k_n is the n-th nonsquarefree number (A013929).

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 4, 1, 3, 2, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 5, 1, 3, 2, 1, 1, 1, 5, 1, 2, 1, 4, 1, 1, 1, 1, 6, 3, 1, 2, 1, 1, 1, 2, 4, 1, 1, 6, 1, 1, 2, 2, 3, 1, 1, 1, 4, 7, 1, 5, 1, 1, 2, 1, 3, 1, 2, 7, 1, 1, 1, 1, 2, 5, 4

Offset: 1

Author

Wesley Ivan Hurt, Jun 09 2020

Keywords

• nonn

Comments

a(n) >= 1.

Examples

			a(4) = 1; The 4th nonsquarefree number, A013929(4) = 12 has 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5) and (6,6) with corresponding products 11, 20, 27, 32, 35, 36. A013929(4) = 12 only divides the product 36, so a(4) = 1.
a(5) = 2; The 5th nonsquarefree number, A013929(5) = 16 has 8 partitions into two parts: (15,1), (14,2), (13,3), (12,4), (11,5), (10,6), (9,7) and (8,8) with corresponding products 15, 28, 39, 48, 55, 60, 63 and 64. A013929(5) = 16 divides two of these products, 48 and 64, so a(5) = 2.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A013929.

Programs

Mathematica

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}] > 0, Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}], {}], {n, 300}] // Flatten