Reciprocals of Odd Numbers adding to 1
Theorem
$105$ is the smallest positive integer $n$ such that $1$ can be expressed as the sum of reciprocals of distinct odd integers such that none are less than $\dfrac 1 n$:
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {33} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {55} + \dfrac 1 {77} + \dfrac 1 {105}$
This sequence is A238795 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
There are $5$ ways of expressing $1$ as the sum of reciprocals of only $9$ distinct odd integers, but then the least term is less than $\dfrac 1 {105}$.
The $9$-term solutions are as follows:
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {35} + \dfrac 1 {45} + \dfrac 1 {231}$
This sequence is A201644 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {231} + \dfrac 1 {315}$
This sequence is A201648 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {33} + \dfrac 1 {45} + \dfrac 1 {385}$
This sequence is A201649 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {165} + \dfrac 1 {693}$
This sequence is A201647 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
- $1 = \dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 9 + \dfrac 1 {11} + \dfrac 1 {15} + \dfrac 1 {21} + \dfrac 1 {135} + \dfrac 1 {10 \, 395}$
This sequence is A201646 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
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Historical Note
David Wells reports in his Curious and Interesting Numbers of $1986$ that this result was reported by Friend Hans Kierstead, Jr. and Harry Lewis Nelson in Volume $10$ of the Journal of Recreational Mathematics, but no information has so far been found to back this up.
It is reported in the various relevant pages in the On-Line Encyclopedia of Integer Sequences that this set was apparently discovered by S. Yamashita in $1976$, but once more corroborative evidence is hard to come by.
Sources
- October 1978: Martin Gardner: Mathematical Games: Puzzles and Number-Theory Problems Arising from the Curious Fractions of Ancient Egypt (Scientific American)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $105$
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Egyptian Fractions: $4$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $105$