A387615 Table of T(b,n) read by ascending antidiagonals, where T(b,n) is the smallest positive integer k for which every nonzero digit in base b appears in the k first multiples of n for b>=2, n>=1.

1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 4, 2, 2, 1, 6, 15, 4, 3, 1, 1, 7, 6, 10, 3, 3, 2, 1, 8, 28, 6, 8, 4, 2, 1, 1, 9, 8, 7, 6, 3, 4, 2, 2, 1, 10, 45, 24, 14, 5, 5, 4, 6, 2, 1, 11, 10, 9, 8, 6, 4, 5, 4, 3, 2, 1, 12, 66, 10, 23, 8, 10, 6, 7, 3, 3, 1, 1, 13, 12, 44, 10, 18, 12, 4, 6, 20, 4, 2, 2, 1

Offset

2,2

Links

Julian Zbigniew Kuryllowicz-Kazmierczak, Table of n, a(n) for n = 2..11326

Formula

T(b,n) = T(b,b*n), T(2,n) = 1, T(b,1) = b-1, T(b,b-1) = ceiling(b/2) for every b>=2, n>=1.

If b and n are coprime and b>=2n-1, T(b,n) = b-1.

If b and n are not coprime and b>n, T(b,n) = ceiling(b(b-1)/n).

Example

T(10,12)=13, because the smallest multiple of 12 containing the digit 5 is 13*12=156, and the rest of the digits shows up in smaller multiples.
Table begins:
       1  2  3  4  5  6  7
  ----------------------
  2 |  1  1  1  1  1  1  1
  3 |  2  2  2  2  1  2  1
  4 |  3  6  2  3  3  2  2
  5 |  4  4  4  3  4  4  4
  6 |  5 15 10  8  3  5  5
  7 |  6  6  6  6  5  4  6

Prog

(PARI) isok(k, b, n) = my(s=[]); for (i=1, k, my(d=digits(i*n, b)); s = setunion(s, Set(d)); ); #select(x->(x>0), s) == b-1;
T(b, n) = my(k=1); while (!isok(k, b, n), k++); k;
matrix(10, 20, b, n, T(b+1, n)) \\ Michel Marcus, Sep 04 2025

Crossrefs

Cf. A202296, A206242, A206243.

Sequence in context: A200154 A208825 A344391 * A089353 A136451 A361894

Adjacent sequences: A387612 A387613 A387614 * A387616 A387617 A387618

Keyword

nonn,tabl,base

Author

Julian Zbigniew Kuryllowicz-Kazmierczak, Sep 03 2025

Status

approved