A333496 Least k of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.
1, 2, 3, 3, 5, 3, 7, 4, 5, 4, 11, 4, 13, 5, 4, 5, 17, 4, 19, 4, 5, 7, 23, 4, 8, 8, 6, 5, 29, 5, 31, 6, 5, 10, 5, 5, 37
Offset: 1
Examples
One of solutions n | [x_1, x_2, ... , x_a(n)] -----+-------------------------------------- 4 | [2, 4, 4] 6 | [2, 3, 6] 8 | [2, 4, 8, 8] 9 | [2, 6, 9, 9, 9] 10 | [2, 5, 5, 10] 12 | [2, 3, 12, 12] 14 | [2, 7, 7, 7, 14] 15 | [2, 3, 10, 15] 16 | [2, 4, 8, 16, 16] 18 | [2, 3, 9, 18] 20 | [2, 4, 5, 20] 21 | [2, 3, 14, 21, 21] 22 | [2, 11, 11, 11, 11, 11, 22] 24 | [2, 3, 8, 24] 25 | [2, 4, 20, 25, 25, 25, 25, 25] 26 | [2, 13, 13, 13, 13, 13, 13, 26] 27 | [2, 3, 18, 27, 27, 27] 28 | [2, 3, 12, 21, 28] 30 | [2, 3, 10, 30, 30] 32 | [2, 3, 16, 24, 32, 32] 33 | [2, 3, 11, 22, 33] 34 | [2, 17, 17, 17, 17, 17, 17, 17, 17, 34] 35 | [2, 3, 14, 15, 35] 36 | [2, 3, 9, 36, 36]
Formula
a(n) <= n.
a(m * n) <= a(n) + m - 1.
If p is prime, a(p) = p.
If m is odd, a(2 * m) <= (m - 1)/2 + 2 because 1 = 1/2 + (m - 1)/2 * 1/m + 1/(2 * m).