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A316935
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a(n)=1; for n > 1, a(n) is the smallest number > n such that the concatenation of all terms from a(1) through a(n) is divisible by the concatenation of the integers 1 through n.
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1
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1, 20, 54, 946, 8180, 93504, 878732, 6841732, 102829509, 19305995230, 1822646098871, 35208071275344, 8691465582891615, 2131922062844429082, 190058192685217102545, 9285111636083665154512, 565278857209893562444229, 49237824030642874847017458, 15301141018410914663693576388
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OFFSET
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1,2
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LINKS
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EXAMPLE
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Let I(n) be the concatenation of the integers 1 through n, and let T(n) be the concatenation of all terms from a(1) through a(n). Then a(n) is the least number which, when appended to T(n-1) -- with leading zeros disallowed -- forms a number T(n) that is strictly greater than I(n) and is a multiple of I(n).
a(1)=1, so T(1)=1. I(2)=12, so a(2)=20 because a(2) is the least number which, when appended to T(1), creates a number T(2) that is greater than I(2)=12 and evenly divisible by it, and the least number greater than 12 that begins with a 1 and is divisible by 12 is 120.
a(2) is not 08 because leading zeros are not allowed.
a(3)=54 because at a(3), I(3)=123, and T(2)=120, which means that T(3) must be 12054, since the least number greater than 123 that is divisible by it but has leading digits 120 is 12054.
Let concat(v) be the concatenation of the elements of vector v. To find a(n) for n >= 3, we find the concatenation of the first n-1 terms. For n = 3, that's 120. We also find the concatenation of the first n positive integers, in this case, 123.
Concatenating 1 gives 1201. 1201 mod 123 = 94 < 123 - 8 so concatenating a 1-digit number without leading zeros doesn't work.
We carry on and compute 12010 mod 123 = 79 >= 123 - 89 so concatenating a 2-digit number without leading zeros works and we stop to find a(3) = 10 + 123 - 79 = 54. (End)
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MATHEMATICA
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getRes[m_, k_, e_] := Module[{}, r = k-Mod[m*10^e, k]; If[r < 10^e && m*10^e+r != k, r += k*(1 + Floor[(10^(e - 1) - r)/k]); If[r>10^e, r=-1], r=-1]; r]; f[m_, k_] := Module[{e=1}, While[(r=getRes[m, k, e])<0, e++]; r]; t[n_] := FromDigits[ Flatten[IntegerDigits[ Range[n]]]]; a[1]=1; a[n_] := a[n] = f[FromDigits[ Flatten[IntegerDigits[Array[a, n-1]]]], t[n]]; Array[a, 20] (* Amiram Eldar, Dec 13 2018 *)
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PROG
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(PARI) first(n) = {my(res = [1, 20]); for(i = 3, n, res = concat(res, [nxt(i, res)])); res}
nxt(n, v) = {my(div = concatnums([1..n]), start = concatnums(v)); start = 10*start+1; f = 1; subt = 8; c = start % div; while(c < div - subt, start *= 10; f*=10; subt = 10*subt + 9; c = start % div); f + div - c}
concatnums(v) = {my(res = v[1]); for(i = 2, #v, res *= 10^#Str(v[i]); res += v[i]); res} \\ David A. Corneth, Dec 18 2018
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CROSSREFS
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Cf. A007908 (concatenation of the numbers from 1 to n).
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KEYWORD
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nonn,easy,base,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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