A367341 - OEIS

%I #85 Jan 09 2025 12:26:45

%S 18,27,36,45,54,63,72,81,918,927,936,945,954,963,972,981,9918,9927,

%T 9936,9945,9954,9963,9972,9981,99918,99927,99936,99945,99954,99963,

%U 99972,99981,999918,999927,999936,999945,999954,999963,999972,999981,9999918,9999927,9999936,9999945,9999954,9999963,9999972,9999981

%N Numbers without comma-successors: these are the numbers k such that if the commas sequence A121805 is started at k instead of 1, there is no second term.

%C Comment from _N. J. A. Sloane_, Nov 19 2023 (Start)

%C Theorem. This sequence consists precisely of the decimal numbers of the form

%C   99...9xy = 100*(10^i-1) + 9*x + 9,

%C with i >= 0 copies of 9, and 1 <= x <= 8.

%C (See link for proof.) This was stated without proof by _David W. Wilson_ in 2007 (see the Angelini link), and was conjectured (in a slightly less precise form) by _Ivan N. Ianakiev_, Nov 16 2023.

%C This implies that the conjecture below is true, as well as the conjecture in A367342.

%C All terms are multiples of 9, and A367342 gives a(n)/9.

%C (End)

%C Numbers k such that A367338(k) = A367339(k) = -1.

%C By definition, A330129 is a subsequence.

%H N. J. A. Sloane, <a href="/A367341/b367341.txt">Table of n, a(n) for n = 1..408</a>

%H Eric Angelini, <a href="/A121805/a121805.pdf">The Commas Sequence</a>, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]

%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>, Fibonacci Quarterly 62:3 (2024), 215-232.

%H 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 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>

%F The first eight terms are given by a(i) = 9*(i+1), for 1 <= i <= 8; thereafter, each successive block of eight terms is obtained by prefixing the terms of the previous block by 9. - _Michael S. Branicky_, Nov 15 2023 [This follows from the theorem above. - _N. J. A. Sloane_, Nov 19 2023]

%p for i from 0 to 4 do t1:=100*(10^i-1);

%p  for x from 1 to 8 do lprint(t1+9*x+9);

%p od: od:

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

%t Select[Range[10^5],fQ[#]&] (* _Ivan N. Ianakiev_, Nov 16 2023 *)

%o (Python)

%o from itertools import islice

%o def ok(n):

%o     an, y = n, 1

%o     while y < 10:

%o         an, y = an + 10*(an%10), 1

%o         while y < 10:

%o             if str(an+y)[0] == str(y):

%o                 an += y

%o                 break

%o             y += 1

%o         if y < 10:

%o             return False

%o     return True

%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Nov 15 2023

%Y Cf. A121805, A367338, A367339, A367340, A367342.

%K nonn,base,changed

%O 1,1

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

%E a(33) and beyond from _Michael S. Branicky_, Nov 15 2023