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Least k such that T(k) includes the consecutive digits of n, where T(k) = k*(k+1)/2.
2

%I #7 Aug 02 2021 12:00:00

%S 0,1,6,2,9,5,3,12,7,13,4,47,15,16,53,5,57,18,60,19,15,6,49,21,69,22,

%T 51,23,7,76,24,21,25,81,68,26,8,27,87,88,28,38,92,29,94,9,30,97,54,31,

%U 75,26,32,17,55,10,33,84,108,34,79,33,72,35,113,30,11,106,183

%N Least k such that T(k) includes the consecutive digits of n, where T(k) = k*(k+1)/2.

%C The T(k) for each n are at A118389.

%H T. D. Noe, <a href="/A118388/b118388.txt">Table of n, a(n) for n = 0..10000</a>

%e ====================

%e n k T(k)

%e ====================

%e 0 0 0

%e 1 1 1

%e 2 6 21

%e 3 2 3

%e 4 9 45

%e 5 5 15

%e 6 3 6

%e 7 12 78

%e 8 7 28

%e 9 13 91

%e 10 4 10

%t nn = 68; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; k = n (n + 1)/2; d = IntegerDigits[k]; s = Sort[FromDigits /@ Flatten[Table[Partition[d, i, 1], {i, Length[d]}], 1]]; i = 1; While[i <= Length[s] && s[[i]] <= nn, If[t[[s[[i]]]] == 0, t[[s[[i]]]] = n; found++]; i++]]; t = Join[{0}, t] (* _T. D. Noe_, Sep 03 2013 *)

%t Module[{tn=Accumulate[Range[200]],trms},trms=Table[SelectFirst[ tn,SequenceCount[ IntegerDigits[ #],IntegerDigits[n]]>0&],{n,70}];Join[ {0},(Sqrt[8#+1]-1)/2&/@trms]] (* _Harvey P. Dale_, Aug 02 2021 *)

%Y Cf. A000217, A118389.

%K base,easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Apr 26 2006

%E Corrected by _T. D. Noe_, Sep 03 2013