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A331786 a(n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

%I A331786 #30 Mar 27 2022 13:39:49
%S A331786 0,2,2,6,8,8,12,14,8,18,38,38,78,98,98,138,158,98,198,398,398,798,998,
%T A331786 998,1398,1598,998,1998,3998,3998,7998,9998,9998,13998,15998,9998,
%U A331786 19998,39998,39998,79998,99998,99998,139998,159998,99998,199998,399998,399998,799998
%N A331786 a(n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.
%C A331786 Write n = 9*s + t, 1 <= t <= 9. The smallest N_0 such that none of S(N_0), S(N_0+1), ..., S(N_0+m-1) is divisible by n is given by N_0 = 10^(u_0) - 10^s*(t-gcd(t,9)+1) + 1, where u_0 is the smallest nonnegative solution to 9*u == -gcd(t,9) (mod n). See A331787 for more detailed information.
%C A331786 From _Bernard Schott_, Mar 25 2022: (Start)
%C A331786 Equivalently, a(n) is the largest number of consecutive integers whose sum of digits (A007953) is never divisible by n (this is the answer to problem of Diophante link).
%C A331786 a(n) ends with 8 when n = 5, 6 and n >= 9 (see formula). (End)
%H A331786 Jianing Song, <a href="/A331786/b331786.txt">Table of n, a(n) for n = 1..1000</a>
%H A331786 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/5026-a389-les-decaxphobes">A389 - Les décaXphobes</a> (in French).
%H A331786 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,10,-10).
%F A331786 If n = 9*s + t, 1 <= t <= 9, then a(n) = 10^s*(2*t-gcd(t,9)+1) - 2. See A331787 for a proof of the formula in base b.
%F A331786 Conjectures from _Colin Barker_, Jan 26 2020: (Start)
%F A331786 G.f.: 2*x^2*(1 + 2*x^2 + x^3 + 2*x^5 + x^6 - 3*x^7 + 5*x^8) / ((1 - x)*(1 - 10*x^9)).
%F A331786 a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n>10.
%F A331786 (End) [This conjecture is correct.]
%F A331786 a(n) = O(10^(n/9)).
%e A331786 The following list gives the smallest example for each 2 <= n <= 27:
%e A331786    2: 9..10 (2)
%e A331786    3: 1..2 (2)
%e A331786    4: 997..1002 (6)
%e A331786    5: 6..13 (8)
%e A331786    6: 7..14 (8)
%e A331786    7: 994..1005 (12)
%e A331786    8: 9999993..10000006 (14)
%e A331786    9: 1..8 (8)
%e A331786   10: 1..18 (18)
%e A331786   11: 999981..1000018 (38)
%e A331786   12: 1..38 (38)
%e A331786   13: 9999999961..10000000038 (78)
%e A331786   14: 951..1048 (98)
%e A331786   15: 961..1058 (98)
%e A331786   16: 9999931..10000068 (138)
%e A331786   17: 999999999999921..1000000000000078 (158)
%e A331786   18: 1..98 (98)
%e A331786   19: 1..198 (198)
%e A331786   20: 99999999801..100000000198 (398)
%e A331786   21: 1..398 (398)
%e A331786   22: 99999999999999601..100000000000000398 (798)
%e A331786   23: 99501..100498 (998)
%e A331786   24: 99601..100598 (998)
%e A331786   25: 99999999301..100000000698 (1398)
%e A331786   26: 99999999999999999999201..100000000000000000000798 (1598)
%e A331786   27: 1..998 (998)
%o A331786 (PARI) a(n) = my(s=(n-1)\9, t=(n-1)%9+1); 10^s*(2*t-gcd(t,9)+1)-2
%Y A331786 Cf. A007953 (S(N)), A051885, A331788.
%Y A331786 Row 10 of A331787.
%K A331786 nonn,base,easy
%O A331786 1,2
%A A331786 _Jianing Song_, Jan 25 2020