A380204 A version of the Josephus problem: a(n) is the surviving integer under the spelling version of the elimination process.

1, 1, 2, 2, 1, 6, 7, 3, 5, 3, 5, 6, 10, 9, 2, 13, 3, 16, 10, 2, 15, 6, 15, 6, 21, 1, 7, 23, 26, 6, 20, 12, 27, 29, 7, 2, 36, 11, 6, 7, 32, 6, 32, 43, 10, 31, 7, 5, 42, 1, 17, 48, 7, 31, 53, 25, 42, 43, 29, 39, 51, 25, 43, 7, 26, 59, 15, 10, 60, 69, 13, 57, 54, 66, 57, 30, 9, 35, 64, 9, 65, 1, 15, 3, 79, 47, 86, 7

Offset: 1

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Jan 16 2025

Keywords

• nonn
• word

Comments

Arrange n people numbered 1, 2, 3, ..., n in a circle, increasing clockwise. Starting with the person numbered 1, spell the letters of O-N-E, moving one person clockwise for each letter. Once you are done, eliminate the next person. Then, spell the letters of T-W-O; in other words, skip three people and eliminate the next person. Following this, spell the letters of T-H-R-E-E; in other words, skip five people and eliminate the next person. Continue until one person remains. The number of this person is a(n).

Examples

			Consider n = 4 people. The first person eliminated is number 4. This leaves the remaining people in the order 1, 2, 3. The second person eliminated is number 1; the people left are in the order 2, 3. The next person eliminated is numbered 3, leaving only the person numbered 2. Thus a(4) = 2.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A005589, A006257, A225381, A321298, A378635, A380201, A380202, A380246, A380247, A380248.

Programs

Python

from num2words import num2words as n2w
def f(n): return sum(1 for c in n2w(n).replace(" and", "") if c.isalpha())
def a(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 1:
        i = (i + f(c))%len(J)
        q = J.pop(i)
        c = c+1
    return J[0]
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Jan 26 2025

Extensions

Terms a(22) and beyond corrected by Michael S. Branicky, Feb 15 2025