Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1
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Theorem
Every positive integer over $77$ can be expressed as the sum of positive integers whose reciprocals add up to $1$.
The full sequence of numbers that cannot be expressed as such is:
- $2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$
This sequence is A051882 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
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Examples
$78$ as Sum of Integers whose Reciprocals total $1$
\(\ds 2 + 6 + 8 + 10 + 12 + 40\) | \(=\) | \(\ds 78\) | ||||||||||||
\(\ds \frac 1 2 + \frac 1 6 + \frac 1 8 + \frac 1 {10} + \frac 1 {12} + \frac 1 {40}\) | \(=\) | \(\ds 1\) |
Sources
- November 1963: R.L. Graham: A theorem on partitions (Journal of the Australian Mathematical Society Vol. 3, no. 4: pp. 435 – 441)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $77$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $77$