A343309 Number of partitions of n into 3 distinct parts x,y,z such that (x+y+z) | x*y*z.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 3, 0, 3, 4, 2, 0, 4, 0, 6, 6, 5, 0, 10, 0, 6, 3, 9, 0, 23, 0, 8, 10, 8, 12, 13, 0, 9, 12, 20, 0, 34, 0, 15, 18, 11, 0, 38, 1, 14, 16, 18, 0, 28, 20, 30, 18, 14, 0, 61, 0, 15, 27, 26, 24, 56, 0, 24, 22, 65, 0, 43, 0, 18, 30, 27, 30, 67, 0, 74

Offset: 1

Author

Wesley Ivan Hurt, Apr 11 2021

Keywords

• nonn

Examples

			a(10) = 2; [1,4,5], [2,3,5], with all parts distinct;
a(12) = 3; [1,3,8], [2,4,6], [3,4,5], with all parts distinct.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A343270.

Programs

Mathematica

Table[Sum[Sum[(1 - KroneckerDelta[i, j]) (1 - KroneckerDelta[n - j, 2 i]) (1 - KroneckerDelta[n - i, 2 j]) (1 - Ceiling[i*j*(n - i - j)/n] + Floor[i*j*(n - i - j)/n]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1 - ceiling(i*j*(n-i-j)/n) + floor(i*j*(n-i-j)/n)) * (1 - [i = j]) * (1 - [n-i = 2*j]) * (1 - [n-j = 2*i]), where [ ] is the Iverson bracket.