A367356 Length of base-3 Commas sequence when started at n.

17, 5, 2, 1, 16, 164, 490, 163, 4, 3, 489, 15, 14, 2, 162, 161, 13, 1472, 488, 1471, 160, 1, 159, 487, 12, 486, 1470, 1469, 11, 158, 157, 1468, 485, 484, 156, 10, 9, 483, 1467, 1466, 8, 155, 154, 1465, 482, 481, 153, 7, 6, 480, 1464, 1463, 5, 41, 152, 40, 1462, 479, 1461, 151, 4, 150, 478, 39, 477, 3, 1460, 2, 38, 149, 37, 1459, 476, 1458, 148, 1, 147, 475, 36, 474, 1457, 1456, 35, 146, 145, 1455, 473, 472, 144, 34, 33, 471, 1454, 1453, 32, 143, 142, 1452, 470, 469

Offset: 1

Author

N. J. A. Sloane, Nov 18 2023

Keywords

• nonn
• base

Comments

a(n) = 1 for n = 4, 22, 76, ... (the numbers 222...2211 in ternary)

We now know that a(n) is finite for all n.

Examples

			For a(1) = 17, see A367355, which has 17 terms.
		

Links

• 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

Crossrefs

Cf. A121805, A330128, A367355.

Programs

Python

from itertools import islice
from sympy.ntheory.factor_ import digits
def a(n, b=3): # generator of terms
    an, y, c = n, 1, 0
    while y < b:
        an, y, c = an + b*(an%b), 1, c+1
        while y < b:
            if str(digits(an+y, b)[1]) == str(y):
                an += y
                break
            y += 1
    return c
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 18 2023