A384774 Elimination order of the first person in a variation of the Josephus problem in which first three people are skipped, then one is eliminated, repeating until all n people are eliminated.

1, 2, 1, 2, 5, 3, 2, 5, 9, 8, 3, 12, 6, 10, 4, 16, 12, 16, 5, 9, 20, 10, 6, 22, 21, 23, 7, 27, 13, 21, 8, 30, 23, 20, 9, 16, 31, 17, 10, 31, 24, 35, 11, 34, 20, 27, 12, 28, 34, 49, 13, 23, 31, 24, 14, 49, 55, 34, 15, 35, 27, 59, 16, 44, 38, 60, 17, 30, 53, 31

Offset: 1

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Jun 09 2025

Keywords

• nonn

Comments

a(4k-1) = k

a(n) = A384770(n,1).

Examples

			Consider n = 4 people. The first person eliminated is number 4. This leaves the remaining people in order 1, 2, 3. On the second step, we eliminate person number 1, implying that the order of elimination of the first person is 2: a(4) = 2.
		

Crossrefs

Cf. First column of A384770.

Cf. A225381, A384772, A088333.

Programs

Maple

A384774 := proc(n::integer)
    local plist,eli,skip,ptr ;
    plist := [seq(i,i=1..n)] ;
    eli :=1 ;
    skip := 3;
    ptr := 0 ;
    while true do
        ptr := modp(ptr+skip,nops(plist)) ;
        if op(ptr+1,plist) = 1 then
            return eli ;
        end if;
        plist := subsop(ptr+1=NULL,plist) ;
        eli := eli+1 ;
    end do:
end proc:
seq(A384774(n),n=1..100) ; # R. J. Mathar, Jul 30 2025

Python

def a(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 0:
        i = (i + 3)%len(J)
        q = J.pop(i)
        if q == 1: return c
        c = c+1
print([a(n) for n in range(1, 71)])