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A333496
Least k of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.
2
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, 11, 6, 5, 41, 4, 43, 6, 5, 13, 47, 5, 10, 6, 6, 6, 53, 5, 6, 5, 7, 16, 59, 5, 61, 17, 6, 7, 6, 6, 67, 7, 7, 5, 71, 5, 73, 20, 6, 6, 6, 5, 79, 6, 7, 22, 83, 5, 6, 23, 8, 5, 89, 5, 5, 7, 9, 25, 6, 5, 97, 8, 5, 5
OFFSET
1,2
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).
For an odd prime p, a(2*p) = floor(p/2) + 2. For prime p>3, a(3*p) = floor(p/6) + 4. - Max Alekseyev, Nov 16 2025
EXAMPLE
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 24 2020
EXTENSIONS
Terms a(38) onward from Max Alekseyev, Nov 16 2025
STATUS
approved