

A069862


Smallest k such that n divides the concatenation of numbers from (n+1) to (n+k), where (n+1) is on the most significant side.


10



1, 2, 2, 2, 5, 2, 9, 4, 8, 10, 10, 8, 22, 16, 5, 4, 2, 8, 3, 20, 20, 10, 17, 12, 25, 22, 26, 16, 25, 20, 110, 20, 11, 2, 20, 8, 998, 52, 38, 20, 60, 20, 4, 32, 35, 42, 50, 20, 96, 50, 2, 96, 93, 26, 10, 20, 3, 50, 44, 20, 46, 40, 45, 40, 50, 32, 86, 32, 17, 20, 75, 72, 26, 926, 50
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OFFSET

1,2


COMMENTS

Minimum number of consecutive subsequent integers after n that must be concatenated together in ascending order such that n divides the concatenated term.
Concatenation always begins at n+1. Note that multiples of 11 seems to require more terms than any other number. 385 requires 9860. 451 requires 100270 terms be concatenated together into a 495,000 digit number.  Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 29 2003; corrected by Chai Wah Wu, Oct 19 2014


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000
C. Seggelin, Concatenation of Consecutive Integers.


EXAMPLE

a(7) = 9 as 7 divides 8910111213141516 the concatenation of numbers from 8(= 7+1) to 16 (= 7+9).
a(5) = 5 because 5 will divide the number formed by concatenating the 5 integers after 5 in ascending order (i.e. 678910). a(385) = 9860 because 385 will divide the concatenation of 386,387,388,...,10245.


MAPLE

c[1] := 1:for n from 2 to 172 do k := 1:g := (n+k) mod n:while(true) do k := k+1:b := convert(n+k, base, 10):g := (g*10^nops(b)+n+k) mod n: if((g mod n)=0) then c[n] := k:break:fi:od:od:seq(c[l], l=1..172);


MATHEMATICA

f[n_] := Block[{k = n + 1}, d = k; While[ !IntegerQ[d/n], k++; d = d*10^Floor[Log[10, k] + 1] + k]; k  n]; Table[ f[n], {n, 1, 75}] (* Robert G. Wilson v, Nov 04 2003 *)


PROG

(Python)
def A069862(n):
....nk, kr, r = n+1, 1, 1 if n > 1 else 0
....while r:
........nk += 1
........kr = (kr + 1) % n
........r = (r*(10**len(str(nk)) % n)+kr) % n
....return nkn # Chai Wah Wu, Oct 20 2014


CROSSREFS

Cf. A069860, A069861, A088797, A088799, A088868, A088870, A088872, A088885.
Records are in A088947, A088343.
Sequence in context: A096396 A029662 A077913 * A075002 A228917 A061311
Adjacent sequences: A069859 A069860 A069861 * A069863 A069864 A069865


KEYWORD

base,nonn


AUTHOR

Amarnath Murthy, Apr 18 2002


EXTENSIONS

More terms from Sascha Kurz, Jan 28 2003


STATUS

approved



