A331789 T(b,n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending antidiagonals.

1, 1, 3, 1, 2, 7, 1, 3, 5, 15, 1, 2, 3, 8, 31, 1, 3, 5, 7, 17, 63, 1, 2, 5, 4, 15, 26, 127, 1, 3, 3, 7, 9, 15, 53, 255, 1, 2, 5, 6, 5, 14, 31, 80, 511, 1, 3, 5, 7, 9, 11, 29, 63, 161, 1023, 1, 2, 3, 4, 9, 6, 23, 24, 63, 242, 2047, 1, 3, 5, 7, 9, 11, 13, 35, 49, 127, 485, 4095

Offset

2,3

Comments

The main sequence is A331787; this is added because some people may search for this.

Links

Jianing Song, Table of n, a(n) for n = 2..7627 (Note: T(b,n) occurs at the ((n+b-2)*(n+b-1)/2-b+3)-th place)

Formula

If n = (b-1)*s + t, 1 <= t <= b-1, then T(b,n) = b^s*(2*t-gcd(t,b-1)+1) - 1. See A331787 for a proof of the formula in base b.

T(b,k) = A331787(b,k) + 1.

T(b,n) = T(b,n-1) + b*T(b,n-b+1) - b*T(b,n-b) for b >= 2, n >= b+1.

T(b,n) = O(b^(n/(b-1))).

Example

Table begins
  b\n  1  2  3   4   5   6    7    8    9    10
   2   1  3  7  15  31  63  127  255  511  1023
   3   1  2  5   8  17  26   53   80  161   242
   4   1  3  3   7  15  15   31   63   63   127
   5   1  2  5   4   9  14   29   24   49    74
   6   1  3  5   7   5  11   23   35   47    35
   7   1  2  3   6   9   6   13   20   27    48
   8   1  3  5   7   9  11    7   15   31    47
   9   1  2  5   4   9  10   13    8   17    26
  10   1  3  3   7   9   9   13   15    9    19

Prog

(PARI) T(b, n) = my(s=(n-1)\(b-1), t=(n-1)%(b-1)+1); b^s*(2*t-gcd(t, b-1)+1)-1

Crossrefs

Cf. A331787.

Cf. A000225 (row 2), A062318 (row 3 with an offset shift), A331788 (row 10).

Sequence in context: A212737 A307078 A134348 * A308001 A191862 A266055

Adjacent sequences: A331786 A331787 A331788 * A331790 A331791 A331792

Keyword

nonn,base,easy,tabl

Author

Jianing Song, Jan 25 2020

Status

approved