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).




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.

$\blacksquare$