A331789 - OEIS

%I #22 Jul 06 2024 19:45:28

%S 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,

%T 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,

%U 2,3,4,9,6,23,24,63,242,2047,1,3,5,7,9,11,13,35,49,127,485,4095

%N 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.

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

%H Jianing Song, <a href="/A331789/b331789.txt">Table of n, a(n) for n = 2..7627</a> (Note: T(b,n) occurs at the ((n+b-2)*(n+b-1)/2-b+3)-th place)

%F 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.

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

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

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

%e Table begins

%e   b\n  1  2  3   4   5   6    7    8    9    10

%e    2   1  3  7  15  31  63  127  255  511  1023

%e    3   1  2  5   8  17  26   53   80  161   242

%e    4   1  3  3   7  15  15   31   63   63   127

%e    5   1  2  5   4   9  14   29   24   49    74

%e    6   1  3  5   7   5  11   23   35   47    35

%e    7   1  2  3   6   9   6   13   20   27    48

%e    8   1  3  5   7   9  11    7   15   31    47

%e    9   1  2  5   4   9  10   13    8   17    26

%e   10   1  3  3   7   9   9   13   15    9    19

%o (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

%Y Cf. A331787.

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

%K nonn,base,easy,tabl

%O 2,3

%A _Jianing Song_, Jan 25 2020