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%I #38 Oct 05 2021 22:47:22
%S 1,1,1,3,1,1,3,5,1,1,9,1,4,7,1,5,23,1,14,1,9,9,13,5,1,21,1,13,12,1,36,
%T 21,9,3,41,1,34,33,9,21,12,9,33,9,1,13,28,5,48,1,23,21,3,1,11,13,14,
%U 41,28,1,114,115,9,41,21,9,23,69,1,61,73,5,14,43,1,145,13,9,127,41,9,95
%N Smallest k such that the concatenation of n consecutive numbers starting with k (from k to n+k-1) is a multiple of n; or 0 if no such number exists.
%C Conjecture: For every n there exists a k.
%C First occurrence of k where a(n)=k: 1, 103, 4, 13, 8, 105, 14, 87, 11, 699, 55, 29, 23, 19, 114, 261, 102, 97, 178, 219, 26, 121, 17, 151, 92, ..., . - _Robert G. Wilson v_
%C a(n)=1 iff n is in A029455. - _Robert G. Wilson v_
%C Increasing a(n)'s: 1, 3, 5, 9, 23, 36, 41, 48, 114, 115, 145, 166, 175, 221, 251, ..., at n = 1, 4, 8, 11, 17, 31, 35, 49, 61, 62, 76, 85, 122, 133, 170, 179, 217, 229, ..., . - _Robert G. Wilson v_
%H David A. Corneth, <a href="/A075001/b075001.txt">Table of n, a(n) for n = 1..10000</a> (first 2500 terms from Robert G. Wilson v)
%e a(11) = 9 as 910111213141516171819 the concatenation of 11 numbers from 9 to 19 is divisible by 11 (11*82737383012865106529).
%t f[n_] := Block[{c = 1, id = Range@n}, While[k = FromDigits@Flatten@IntegerDigits@id/n; ! IntegerQ@k, id++; c++ ]; c]; Array[f, 82] (* _Robert G. Wilson v_, Oct 20 2007 *)
%o (PARI) /* The following program assumes the conjecture is true. */ /* It has found nonzero a(n) for n up to 500. */ {for(n=1,500, k=0; until(s%n==0,k++; s=0; for(m=k,k+n-1, s=s*(10^length(Str(m)))+m)); print1(k,","))}
%o (PARI) a(n) = {my(ld = 1, hd = n, qd, m = Mod(1, n), pow10, qdn = #digits(n), t=log(10*n+.5)\log(10)); qd = n*t+t-10^t\9; pow10 = Mod(10, n)^(qd-1); for(i = 2, n, m = m * Mod(10, n)^#digits(i) + i; ); while(1, if(lift(m) == 0, return(ld)); m -= ld * pow10; hd++; m = m * Mod(10, n)^#digits(hd) + hd; ld++; pow10*=10^(#digits(hd) - #digits(ld)); ) } \\ _David A. Corneth_, Aug 23 2020
%Y Cf. A058183, A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075000, A077306, A075000.
%Y Cf. A029455 (indices of 1's).
%K nonn,base
%O 1,4
%A _Amarnath Murthy_, Aug 31 2002
%E More terms from _Rick L. Shepherd_, Sep 03 2002
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