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

0,1

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)

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).

Example

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

Prog

(PARI) A(n, k) = { for (b = 2, oo, if (sumdigits(n*k, b) == sumdigits(n, b) * sumdigits(k, b), return (b); ); ); }

Crossrefs

Cf. A319478, A362366.

Sequence in context: A287091 A204551 A292563 * A211661 A111972 A073458

Adjacent sequences: A362364 A362365 A362366 * A362368 A362369 A362370

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Apr 17 2023

Status

approved