A367355 -id:A367355 - cp's OEIS frontend

A367604 Length of commas sequence (cf. A121805) if start at 1 and do the calculations in base n; or -1 if the sequence is infinite.

-1, 17, 2, 1259, 3243760, 33779, 8367, 3, 2137453, 29347, 4, 27, 3097837317, 75455289096, 144693554136426354, 586, 8250248375768635503445567685, 4956282, 51496560713, 7977231, 4, 560002, 48, 779641, 10712620148411, 44948082868036315658034347512222651

Offset: 2

N. J. A. Sloane, Dec 08 2023

Cf. A121805, A367605 (last term).

The sequences for bases 3, 8, and 10 are A367355, A367344, and A121805.

More terms from Michael S. Branicky, Dec 08 2023

A367621 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.

Original entry on oeis.org

1, 5, 12, 13, 18, 20, 27, 28, 32, 39, 40, 44, 51, 52, 57, 59, 67, 72, 74, 81, 82, 86, 93, 94, 98, 105, 106, 110, 117, 118, 122, 129, 130, 134, 141, 142, 146, 153, 154, 158, 166, 171, 173, 181, 186, 188, 196, 201, 203, 211, 216, 218, 226, 231, 233, 241, 245, 252

Offset: 1

Michael S. Branicky and N. J. A. Sloane, Dec 20 2023

Analogous to A367620, but with comma-children computed in base 3 (terms are shown in base 10, however).
We know from A367619 that the comma-child graph in base 3, starting at 1, is an infinite tree rooted at 1. By König's Infinity Lemma, an infinite path in that graph exists and hence this sequence is well defined for all n.  Therefore, at any bifurcation point, one or both forks will extend to infinity. The definition of this sequence requires that we choose the smallest fork that has an infinite continuation.
The terms in the data and b-file include a number of bifurcation points, but in each case the path chosen was the only one that did not lead to a finite sequence; see linked a-file.
We conjecture that choosing down-up-down-up-... is an infinite path, visiting the base-3 terms 1 2^{1+4*j} then 2 0^{2+4*j} for j in 0..oo, where ^ denotes repeated concatenation. This has been tested empirically up to j = 4300.

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
Michael S. Branicky, Bifurcation points on the road to infinity in the base-3 comma-child graph, starting at 1
Giovanni Resta, Graphical representation of a portion of the graph

Cf. A367355, A367618, A367619, A367620.

A367623 Number of comma-children of n in base 3.

Original entry on oeis.org

2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 2023

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

Cf. A121805, A367355, A367356, A367606-A367609, A367622.

from sympy.ntheory.factor_ import digits
def a(k, base=3):
    m = k + base*(k%base)
    return len([m+y for y in range(1, base) if digits(m+y, base)[1] == y])
print([a(n) for n in range(1, 96)]) # Michael S. Branicky, Dec 23 2023

cp's OEIS Frontend

A367604 Length of commas sequence (cf. A121805) if start at 1 and do the calculations in base n; or -1 if the sequence is infinite.

Views

Author

Keywords

Crossrefs

Extensions

A367621 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.

Views

Author

Keywords

Comments

Links

Crossrefs

A367623 Number of comma-children of n in base 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Python