A333437 Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.
1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 3, 2, 1, 0, 0, 0, 0, 2, 2, 3, 2, 1, 0, 0, 0, 1, 3, 6, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1
Offset: 1
Examples
1 = 1/2 + 1/6 + 1/6 + 1/6 = 1/3 + 1/3 + 1/6 + 1/6 = 1/3 + 1/4 + 1/4 + 1/6. So T(6,4) = 3. Triangle begins: n\k | 1 2 3 4 5 6 7 8 9 10 11 12 -----+---------------------------------------- 1 | 1; 2 | 0, 1; 3 | 0, 0, 1; 4 | 0, 0, 1, 1; 5 | 0, 0, 0, 0, 1; 6 | 0, 0, 1, 3, 2, 1; 7 | 0, 0, 0, 0, 0, 0, 1; 8 | 0, 0, 0, 1, 3, 3, 2, 1; 9 | 0, 0, 0, 0, 2, 2, 3, 2, 1; 10 | 0, 0, 0, 1, 3, 6, 7, 5, 3, 1; 11 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 12 | 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1;
Formula
T(n,n) = 1.
If n is prime, T(n,k) = 0 for 1 <= k < n.