OFFSET
1,3
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).
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..20000
Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See p. 17.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,word
AUTHOR
Tanya Khovanova and the MIT PRIMES STEP junior group, Jan 16 2025
EXTENSIONS
Terms a(22) and beyond corrected by Michael S. Branicky, Feb 15 2025
STATUS
approved