A367346 - OEIS

%I #41 Feb 07 2024 13:21:43

%S 14,33,52,71,118,227,336,445,554,663,772,881,1918,2927,3936,4945,5954,

%T 6963,7972,8981,19918,29927,39936,49945,59954,69963,79972,89981,

%U 199918,299927,399936,499945,599954,699963,799972,899981,1999918,2999927,3999936,4999945,5999954,6999963,7999972,8999981

%N Numbers k such that there is more than one possible solution for A367338(k).

%C The number of solutions is either 0, 1, or 2.

%C The definition of A121805 instructs us to pick the smallest solution, so there is no ambiguity in the definition of A121805. The present sequence shows that there are very few cases where there is any possible ambiguity.

%C The sequence begins with the four exceptional terms 14, 33, 52, 71. It also includes all numbers with decimal expansions of the form d 9^i d (9-d), where juxtaposition is concatenation, ^ denotes repeated concatenation of digits, 1 <= d <= 8, and i >= 0, with associated next terms in the commas sequence being either d 9^(i+2) or (d+1) 0^(i+2). It is conjectured that there are no other terms. - _Michael S. Branicky_, Nov 16 2023

%C The conjecture is true; see link. - _Michael S. Branicky_, Nov 21 2023

%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Youtube</a>

%e In the commas sequence starting at 14, the next term could be either 59 or 60, because both 14,59 and 14,60 satisfy the "commas" rule (since both 14 + 45 = 59 and 14 + 46 = 60).

%t fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]>1;

%t Select[Range[10^6],fQ[#]&] (* _Ivan N. Ianakiev_, Dec 16 2023 *)

%Y Cf. A121805, A367338, A367341.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Nov 15 2023

%E a(30) and beyond from _Michael S. Branicky_, Nov 16 2023

%E Second comment edited by _N. J. A. Sloane_, Nov 20 2023