Smallest Sum of 2 Lucky Numbers in n Ways
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Sequence
The sequence of positive integers which can be expressed as the sum of $2$ distinct lucky numbers in $n$ different ways begins:
- $4, 10, 16, 34, 46, 144, 76, 112, 100, 148, 166, 136, 202, 226, 238, 268, 298, 304, 310, 352, 400, 430, 490, \ldots$
This sequence is A333904 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
This article, or a section of it, needs explaining. In particular: Please note that the above sequence is still at draft stage, having been raised as a new sequence by Matt Westwood on $9$th April $2020$. In due course it should appear in the OEIS. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code. |
Proof
The technique is to create all sums of $2$ distinct lucky numbers and count the number of times each sum occurs.
To reach the term $46$, for example, is it sufficient to go as high as building the sums up to the lucky number immediately less than $46$.
The sequence of lucky numbers begins:
- $1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, \ldots$
and so:
\(\ds 4\) | \(=\) | \(\ds 1 + 3\) | ||||||||||||
\(\ds 8\) | \(=\) | \(\ds 1 + 7\) | ||||||||||||
\(\ds 10\) | \(=\) | \(\ds 1 + 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 7\) | ||||||||||||
\(\ds 12\) | \(=\) | \(\ds 3 + 9\) | ||||||||||||
\(\ds 14\) | \(=\) | \(\ds 1 + 13\) | ||||||||||||
\(\ds 16\) | \(=\) | \(\ds 1 + 15\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 13\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 9\) | ||||||||||||
\(\ds 18\) | \(=\) | \(\ds 3 + 15\) | ||||||||||||
\(\ds 20\) | \(=\) | \(\ds 7 + 13\) | ||||||||||||
\(\ds 22\) | \(=\) | \(\ds 1 + 21\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 15\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 13\) | ||||||||||||
\(\ds 24\) | \(=\) | \(\ds 3 + 21\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 15\) | ||||||||||||
\(\ds 26\) | \(=\) | \(\ds 1 + 25\) | ||||||||||||
\(\ds 28\) | \(=\) | \(\ds 3 + 25\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 21\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 15\) | ||||||||||||
\(\ds 30\) | \(=\) | \(\ds 9 + 21\) | ||||||||||||
\(\ds 32\) | \(=\) | \(\ds 1 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 25\) | ||||||||||||
\(\ds 34\) | \(=\) | \(\ds 1 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 25\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 21\) | ||||||||||||
\(\ds 36\) | \(=\) | \(\ds 3 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 21\) | ||||||||||||
\(\ds 38\) | \(=\) | \(\ds 1 + 37\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 25\) | ||||||||||||
\(\ds 40\) | \(=\) | \(\ds 3 + 37\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 25\) | ||||||||||||
\(\ds 42\) | \(=\) | \(\ds 9 + 33\) | ||||||||||||
\(\ds 44\) | \(=\) | \(\ds 1 + 43\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 37\) | ||||||||||||
\(\ds 46\) | \(=\) | \(\ds 3 + 43\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 37\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 13 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 25\) | ||||||||||||
\(\ds 48\) | \(=\) | \(\ds 15 + 33\) | ||||||||||||
\(\ds 50\) | \(=\) | \(\ds 1 + 49\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 43\) | ||||||||||||
\(\ds 52\) | \(=\) | \(\ds 1 + 51\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 49\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 31\) | ||||||||||||
\(\ds 54\) | \(=\) | \(\ds 3 + 51\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 21 + 33\) | ||||||||||||
\(\ds 56\) | \(=\) | \(\ds 25 + 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 49\) | ||||||||||||
\(\ds 58\) | \(=\) | \(\ds 25 + 33\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 + 51\) | ||||||||||||
\(\ds 64\) | \(=\) | \(\ds 1 + 63\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 31 + 33\) | ||||||||||||
\(\ds 66\) | \(=\) | \(\ds 3 + 63\) | ||||||||||||
\(\ds 70\) | \(=\) | \(\ds 7 + 63\) |
The first $5$ terms of the sequence are seen to appear.
$\blacksquare$