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

14, 33, 52, 71, 118, 227, 336, 445, 554, 663, 772, 881, 1918, 2927, 3936, 4945, 5954, 6963, 7972, 8981, 19918, 29927, 39936, 49945, 59954, 69963, 79972, 89981, 199918, 299927, 399936, 499945, 599954, 699963, 799972, 899981, 1999918, 2999927, 3999936, 4999945, 5999954, 6999963, 7999972, 8999981

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

Table of n, a(n) for n=1..44.

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

Cf. A121805, A367338, A367341.

Sequence in context: A191676 A191866 A162279 * A090090 A116150 A019272

Adjacent sequences: A367343 A367344 A367345 * A367347 A367348 A367349

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