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.
a(2)=22 because 22 is a factor of 21201918171615141312.
a(7) = 2 because 7 will divide the number formed by concatenating the 2 integers prior to 7 in ascending order (i.e. 56). a(6) = 0 because 6 will not divide 5, 45, 345, 2345, 12345, or 012345.
concat(2857141, 2857142) / 2857143 = 28571412857142 / 2857143 = 9999994.
with(numtheory): P:=proc(q) local c,n; for n from 1 to q do c:=n*10^(ilog10(n+1)+1)+n+1; if type(c/(n+2),integer) then print(n); fi; od; end: P(10^9);
Select[Range[10^7], Divisible[FromDigits@ Flatten@ Map[IntegerDigits, {#, # + 1}], # + 2] &] (* Michael De Vlieger, Jan 19 2017 *)
isok(n) = !(eval(Str(n, n+1)) % (n+2)); \\ Michel Marcus, Oct 09 2018
a(4) = 6 because concat(2, 3, 4, 5, 6, 7, 8, 9, 10) = 2345678910 is a multiple of 6 and 6 is the least number to have this property.
P:=proc(q) local a,j,k,n; for k from 1 by 2 to q do for n from 1 to q do a:=n; for j from 1 to k-1 do a:=a*10^(ilog10(n+j)+1)+n+j; od; if type(a/(n+(k-1)/2),integer) then print(n+(k-1)/2); break; fi; od; od; end: P(10^9);
Table[SelectFirst[Range[10^3], Function[k, If[And[FreeQ[#, ?(# < 0 &)], First@ # != 0], Divisible[FromDigits@ Flatten@ Map[IntegerDigits, #], k]] &[k + Range[-n, n]] ] ], {n, 0, 64}] (* _Michael De Vlieger, Sep 27 2017 *)
isok(k, n) = {my(s = ""); for (i=0, 2*n, s = concat(s, k-n+i);); (eval(s) % k) == 0;}
a(n) = {my(k = n+1); while(!isok(k,n), k++); k;} \\ Michel Marcus, Oct 06 2017
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