OFFSET
1,1
COMMENTS
The number of solutions is either 0, 1, or 2.
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.
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
The conjecture is true; see link. - Michael S. Branicky, Nov 21 2023
LINKS
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
EXAMPLE
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).
MATHEMATICA
fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]}, Length[Select[k, #-n==FromDigits[{Last[IntegerDigits[n]], First[IntegerDigits[#]]}]&]]]>1;
Select[Range[10^6], fQ[#]&] (* Ivan N. Ianakiev, Dec 16 2023 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Nov 15 2023
EXTENSIONS
a(30) and beyond from Michael S. Branicky, Nov 16 2023
Second comment edited by N. J. A. Sloane, Nov 20 2023
STATUS
approved