A367341 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.

18, 27, 36, 45, 54, 63, 72, 81, 918, 927, 936, 945, 954, 963, 972, 981, 9918, 9927, 9936, 9945, 9954, 9963, 9972, 9981, 99918, 99927, 99936, 99945, 99954, 99963, 99972, 99981, 999918, 999927, 999936, 999945, 999954, 999963, 999972, 999981, 9999918, 9999927, 9999936, 9999945, 9999954, 9999963, 9999972, 9999981

Offset: 1

Author

N. J. A. Sloane, Nov 15 2023

Keywords

• nonn
• base

Comments

Comment from N. J. A. Sloane, Nov 19 2023 (Start)

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

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

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

(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.

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

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

(End)

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

By definition, A330129 is a subsequence.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..408
• Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
• 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, Fibonacci Quarterly 62:3 (2024), 215-232.
• Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
• N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Crossrefs

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

Programs

Maple

for i from 0 to 4 do t1:=100*(10^i-1);
 for x from 1 to 8 do lprint(t1+9*x+9);
od: od:

Mathematica

fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]}, Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]] =={};
Select[Range[10^5],fQ[#]&] (* Ivan N. Ianakiev, Nov 16 2023 *)

Python

from itertools import islice
def ok(n):
    an, y = n, 1
    while y < 10:
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
        if y < 10:
            return False
    return True
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Nov 15 2023

Formula

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]

Extensions

a(33) and beyond from Michael S. Branicky, Nov 15 2023