A362367 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) is the least base >= 2 where the product n * k can be computed without carry.
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 2, 2, 2, 2, 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 2 2 2 2 2 2 2 2 2 2 2 2
2 | 2 2 2 2 2 2 2 2 2 2 2 2 2
3 | 2 2 2 3 2 2 3 3 2 2 2 3 3
4 | 2 2 2 2 2 2 2 2 2 2 2 2 2
5 | 2 2 2 2 2 4 2 5 2 2 3 5 2
6 | 2 2 2 3 2 2 5 5 2 2 2 5 3
7 | 2 2 2 3 2 5 5 5 2 2 3 7 6
8 | 2 2 2 2 2 2 2 2 2 2 2 2 2
9 | 2 2 2 2 2 2 2 2 2 3 2 3 2
10 | 2 2 2 2 2 3 2 3 2 2 3 5 2
11 | 2 2 2 3 2 5 5 7 2 3 5 5 3
12 | 2 2 2 3 2 2 3 6 2 2 2 3 3
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..10010
- 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););); }
Formula
A(n, k) <= max(2, n*k + 1).
A(n, k) = A(k, n).
A(n, 0) = A(n, 1) = A(n, 2) = 2.
A(n, n) = A319478(n).