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A367346 Numbers k such that there is more than one possible solution for A367338(k).

%I A367346 #43 Jan 09 2025 12:33:37
%S A367346 14,33,52,71,118,227,336,445,554,663,772,881,1918,2927,3936,4945,5954,
%T A367346 6963,7972,8981,19918,29927,39936,49945,59954,69963,79972,89981,
%U A367346 199918,299927,399936,499945,599954,699963,799972,899981,1999918,2999927,3999936,4999945,5999954,6999963,7999972,8999981
%N A367346 Numbers k such that there is more than one possible solution for A367338(k).
%C A367346 The number of solutions is either 0, 1, or 2.
%C A367346 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 A367346 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 A367346 The conjecture is true; see link. - _Michael S. Branicky_, Nov 21 2023
%H A367346 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>, Fibonacci Quarterly 62:3 (2024), 215-232.
%H A367346 Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, <a href="/A121805/a121805_1.pdf">The Comma Sequence: A Simple Sequence With Bizarre Properties</a>, Local copy.
%H A367346 N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Eric Angelini's Comma Sequence</a>, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane2024.pdf">Slides</a>
%e A367346 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 A367346 fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]>1;
%t A367346 Select[Range[10^6],fQ[#]&] (* _Ivan N. Ianakiev_, Dec 16 2023 *)
%Y A367346 Cf. A121805, A367338, A367341.
%K A367346 nonn,base
%O A367346 1,1
%A A367346 _N. J. A. Sloane_, Nov 15 2023
%E A367346 a(30) and beyond from _Michael S. Branicky_, Nov 16 2023
%E A367346 Second comment edited by _N. J. A. Sloane_, Nov 20 2023