A367339 -id:A367339 - cp's OEIS frontend

A367338 Comma-successor to n: second term of commas sequence if initial term is n, or -1 if there is no second term.

12, 24, 36, 48, 61, 73, 85, 97, 100, 11, 23, 35, 47, 59, 72, 84, 96, -1, 110, 22, 34, 46, 58, 71, 83, 95, -1, 109, 120, 33, 45, 57, 69, 82, 94, -1, 108, 119, 130, 44, 56, 68, 81, 93, -1, 107, 118, 129, 140, 55, 67, 79, 92, -1, 106, 117, 128, 139, 150, 66, 78, 91, -1, 105, 116

Offset: 1

N. J. A. Sloane, Nov 15 2023

Construct the commas sequence as in A121805, but take the first term to be n. Then a(n), the comma-successor to n, is the second term, or -1 if no second term exists.
More generally, we define a comma-child of n to be any number m with the property that m-n = 10*x+y, where x is the least significant digit of n and y is the most significant digit of m.
A positive number can have 0, 1, or 2 comma-children. In accordance with the Law of Primogeniture, the first-born child (i.e. the smallest), if there is one, is the comma-successor.
Comment from N. J. A. Sloane, Nov 19 2023: (Start)
The following is a proof of a slight modification of a conjecture made by Ivan N. Ianakiev in A367341.
The Comma-Successor Theorem.
Let D(b) denote the set of numbers k which have no comma-successor in base b ("comma-successor" is the base-b generalization of the rule that defines A121805). If a commas sequence reaches a number in D(b) it will end there.
Then D(b) consists precisely of the numbers which when written in base b have the form
  cc...cxy = (b^i-1)*b^2/(b-1) + b*x + y,
   with i >= 0 copies of c = b-1, where x and y are in the range [1..b-2] and satisfy x+y = b-1.  .... (*)
For b = 10 the numbers D(10) are listed in A367341.
For an outline of the proof, see the attached text-file.
Note that in base b = 2, no values of x satisfying (*) exist, and the theorem asserts that D(2) is empty. In fact it is easy to check directly that every commas sequence in base 2 is infinite.  If the initial term is 0 or 1 mod 4 then the sequence will merge with A042948, and if the initial term is 2 or 3 mod 4 then the sequence will merge with A042964.
(End)

			a(1) = A121803(2) = 12,
a(2) = A139284(2) = 24,
a(3) = 36, since the full commas sequence starting with 3 is [3, 36] (which also implies a(36) = -1),
a(4) = A366492(2) = 48, and so on.
60 is the first number that is a comma-child (a member of A367312) but is missing from the present sequence (it is a comma-child but not a comma-successor, since it loses out to 59).

Michael S. Branicky, Table of n, a(n) for n = 1..10000
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

A367346 lists those n for which there is more than one choice for the second term.
A367612 lists the numbers that are comma-children of some number k.
Cf. A121805, A139284, A366492, A367337.
See also A042948, A042964, A367339, A367340, A367341.

Ldigit:=proc(n) local v; v:=convert(n, base, 10); v[-1]; end;
A367338 := proc(n) local f,i,d;
f := (n mod 10);
d:=10*f;
for i from 1 to 9 do
d := d+1;
if Ldigit(n+d) = i then return(n+d); fi;
od:
return(-1);
end;
for n from 1 to 50 do lprint(n, A367338(n)); od: # N. J. A. Sloane, Dec 06 2023

a[n_] := a[n] = Module[{l = n, y = 1, d}, While[y < 10, l = l + 10*(Mod[l, 10]); y = 1; While[y < 10, d = IntegerDigits[l + y][[1]]; If[d == y, l = l + y; Break[];]; y++;]; If[y < 10, Return[l]];]; Return[-1];];
Table[a[n], {n, 1, 65}] (* Robert P. P. McKone, Dec 18 2023 *)

from itertools import islice
def a(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 an
    return -1
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Nov 15 2023

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.

Original entry on oeis.org

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

N. J. A. Sloane, Nov 15 2023

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.

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

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

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:

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 *)

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

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]

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

A367340 List of distinct (positive) numbers appearing in A367338.

Original entry on oeis.org

11, 12, 22, 23, 24, 33, 34, 35, 36, 44, 45, 46, 47, 48, 55, 56, 57, 58, 59, 61, 66, 67, 68, 69, 71, 72, 73, 77, 78, 79, 81, 82, 83, 84, 85, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118

Offset: 1

N. J. A. Sloane, Nov 15 2023

This is the list of comma-successors.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
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

Cf. A121805, A367338, A367339, A367341.

def cp(n):
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    kk = k + 10*x + y-1
    return k if k > 0 and int(str(kk)[0]) != y-1 else -1
def ok(n): return n > 0 and cp(n) > 0
print([k for k in range(1, 119) if ok(k)]) # Michael S. Branicky, Dec 18 2023

A367609 Comma-number associated with A367607(n), and written in base 3, or -1 if A367607(n) = -1.

Original entry on oeis.org

11, 21, 1, -1, 21, 2, 11, 21, 1, 11, 22, 1, 11, 22, 1, 12, 22, 2, 12, 21, 2, -1, 21, 2, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 22, 1, 11, 22, 1, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 22, 2, 12, 21, 2, -1, 21, 2, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11, 21, 1, 11

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is a base-3 analog of A367339.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
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

Cf. A121085, A367339, A367355, A367356, A367606-A367608.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((int("".join(map(str, digits(b+y-n, 3)[1:]))) for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Dec 11 2023

A367608 Comma-number associated with A367606(n), but written in base 10, or -1 if A367606(n) = -1.

Original entry on oeis.org

4, 7, 1, -1, 7, 2, 4, 7, 1, 4, 8, 1, 4, 8, 1, 5, 8, 2, 5, 7, 2, -1, 7, 2, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 8, 1, 4, 8, 1, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 8, 2, 5, 7, 2, -1, 7, 2, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4

Offset: 1

N. J. A. Sloane, Dec 11 2023

If n has a comma-successor m (say) in base 3, then a(n) is the comma-number linking n and m, and is equal to m-n; a(n) = -1 if n has no successor. See A367338 for definitions.

This is a base-3 analog of A367339.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A121085, A367339, A367355, A367356, A367606-A367609.

from sympy.ntheory.factor_ import digits
def a(n):
    b = n + 3*(n%3)
    return next((b+y-n for y in [1, 2] if digits(b+y, 3)[1] == y), -1)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Dec 11 2023

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A367338 Comma-successor to n: second term of commas sequence if initial term is n, or -1 if there is no second term.

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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.

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A367340 List of distinct (positive) numbers appearing in A367338.

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A367609 Comma-number associated with A367607(n), and written in base 3, or -1 if A367607(n) = -1.

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A367608 Comma-number associated with A367606(n), but written in base 10, or -1 if A367606(n) = -1.

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A368365 Numbers n whose comma-number is greater than n.

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