Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1

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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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): $77$