This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331789 #22 Jul 06 2024 19:45:28 %S A331789 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 A331789 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 A331789 2,3,4,9,6,23,24,63,242,2047,1,3,5,7,9,11,13,35,49,127,485,4095 %N 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. %C A331789 The main sequence is A331787; this is added because some people may search for this. %H A331789 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 A331789 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 A331789 T(b,k) = A331787(b,k) + 1. %F A331789 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 A331789 T(b,n) = O(b^(n/(b-1))). %e A331789 Table begins %e A331789 b\n 1 2 3 4 5 6 7 8 9 10 %e A331789 2 1 3 7 15 31 63 127 255 511 1023 %e A331789 3 1 2 5 8 17 26 53 80 161 242 %e A331789 4 1 3 3 7 15 15 31 63 63 127 %e A331789 5 1 2 5 4 9 14 29 24 49 74 %e A331789 6 1 3 5 7 5 11 23 35 47 35 %e A331789 7 1 2 3 6 9 6 13 20 27 48 %e A331789 8 1 3 5 7 9 11 7 15 31 47 %e A331789 9 1 2 5 4 9 10 13 8 17 26 %e A331789 10 1 3 3 7 9 9 13 15 9 19 %o A331789 (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 A331789 Cf. A331787. %Y A331789 Cf. A000225 (row 2), A062318 (row 3 with an offset shift), A331788 (row 10). %K A331789 nonn,base,easy,tabl %O A331789 2,3 %A A331789 _Jianing Song_, Jan 25 2020