A367312 -id:A367312 - 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

A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

Original entry on oeis.org

0, 6, 4, 1, 3, 9, 2, 8, 2, 0, 6, 4, 2, 5, 7, 1, 6, 8, 4, 2, 2, 0, 8, 8, 7, 1, 6, 5, 1, 2, 7, 1, 8, 1, 6, 8, 7, 3, 9, 3, 6, 5, 6, 8, 2, 8, 4, 4, 6, 4, 6, 4, 0, 1, 3, 9, 5, 5, 9, 5, 7, 7, 0, 0, 2, 2, 5, 2, 5, 7, 6, 2, 7, 9, 8, 3, 6, 9, 3, 2, 1, 7, 2, 4, 9, 4, 7

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .

Cf. A113024, A367309, A367312.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

One initial 0 inserted by Artur Jasinski, Aug 04 2025

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.

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