%I #27 Aug 01 2019 10:36:25
%S 1,11,12,112,113,132,1114,1133,1143,11134,11144,11154,11443,111155,
%T 111165,111544,111554,256318,1111555,1111655,1111665,1115554,1194332,
%U 11111766,11111776,11116565,11116665,12337741,12377441,111116766,111117676,111117776,111166665,113777242,123377741,123777441
%N Smallest integer N such that all numbers from 1 to n (n being N's rank in the sequence) can be built by the sum of contiguous digits of N.
%C The 18th term (256318) is peculiar: it is the only one among the first 43 terms of the sequence that doesn't start with 1 and it is the only one having a digit-sum (25) larger than its rank in the sequence (18).
%C Barry Schwarz has computed a(44) to a(59), the latter being the second term of the sequence not to start with 1; a(59) = 213388888552. Barry estimates that a(60) probably has at least 13 digits.
%C a(60) = 1111111999998 has indeed 13 digits. a(100) <= 11123999999999461. - _Giovanni Resta_, Aug 01 2019
%H Giovanni Resta, <a href="/A296447/b296447.txt">Table of n, a(n) for n = 1..85</a> (terms 1..59 from Jean-Marc Falcoz and Barry Schwarz)
%e The 18th term of the sequence is 256318. The 18 numbers from 1 to 18 can be built summing a certain set of contiguous digits of 256318: 1 is the digit 1; 2 is the digit 2; 3 is the digit 3; 4 is 3+1 (contiguous); 5 is the digit 5; 6 is the digit 6; 7 is 2+5 (contiguous); 8 is the digit 8; 9 is 6+3 (contiguous); 10 is 6+3+1 (contiguous); 11 is 5+6 (contiguous); 12 is 3+1+8 (contiguous); 13 is 2+5+6 (contiguous); 14 is 5+6+3 (contiguous); 15 is 5+6+3+1 (contiguous); 16 is 2+5+6+3 (contiguous); 17 is 2+5+6+3+1 (contiguous); 18 is 6+3+1+8 (contiguous).
%t Array[With[{r = Range@ #}, SelectFirst[Range[10^6], SequenceCount[Union@ Map[Total, #] &@ Apply[Join, Table[Partition[#, i, 1], {i, Length@#}]] &@ IntegerDigits@ #, r] == 1 &]] &, 18] (* or *)
%t With[{s = Array[LengthWhile[#, # == 1 &] &@ Differences@ Union@ Map[Total, #] &@ Apply[Join, Table[Partition[#, i, 1], {i, Length@ #}]] &@ IntegerDigits@ # &, 10^6]}, SelectFirst[#, FreeQ[IntegerDigits@#, 0] &] & /@ Values@ PositionIndex@ s] (* _Michael De Vlieger_, Dec 13 2017 *)
%K nonn,base
%O 1,2
%A _Eric Angelini_ and _Jean-Marc Falcoz_, Dec 13 2017