A362366 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the least base >= 2 where the sum n + k can be computed without carry.
2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 5, 3, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 5, 5, 4, 4, 5, 5, 2, 2, 2, 3, 2, 6, 4, 4, 4, 6, 2, 3, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 5, 5, 3, 2, 5, 5, 5, 2, 3, 5, 5, 2
Offset: 0
Examples
Array A(n, k) begins:
n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12
----+-----------------------------------------
0 | 2 2 2 2 2 2 2 2 2 2 2 2 2
1 | 2 3 2 3 2 4 2 3 2 3 2 5 2
2 | 2 2 5 3 2 2 3 5 2 2 5 5 2
3 | 2 3 3 3 2 3 5 6 2 3 3 3 2
4 | 2 2 2 2 3 4 4 4 2 2 2 2 3
5 | 2 4 2 3 4 4 4 5 2 3 2 5 3
6 | 2 2 3 5 4 4 5 5 2 2 3 3 5
7 | 2 3 5 6 4 5 5 5 2 3 3 5 5
8 | 2 2 2 2 2 2 2 2 6 3 5 5 6
9 | 2 3 2 3 2 3 2 3 3 3 3 3 3
10 | 2 2 5 3 2 2 3 3 5 3 3 5 3
11 | 2 5 5 3 2 5 3 5 5 3 5 5 3
12 | 2 2 2 2 3 3 5 5 6 3 3 3 3
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10010
- Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
- Rémy Sigrist, Colored representation of the array for n, k <= 1024 (the color is function of A(n, k), black pixels denote 2's)
Programs
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PARI
A(n, k) = { for (b = 2, oo, if (sumdigits(n+k, b) == sumdigits(n, b) + sumdigits(k, b), return (b););); }