A381591 Elimination order of the first person in a variation of the Josephus problem, where there are n people total and two people are skipped each time.

1, 1, 2, 4, 2, 6, 6, 3, 9, 6, 4, 7, 11, 5, 11, 15, 6, 13, 11, 7, 12, 16, 8, 23, 18, 9, 22, 16, 10, 17, 31, 11, 27, 30, 12, 35, 21, 13, 22, 37, 14, 30, 35, 15, 32, 26, 16, 27, 35, 17, 47, 37, 18, 53, 31, 19, 32, 47, 20, 57, 56, 21, 51, 36, 22, 37, 65, 23, 49, 70

Offset: 1

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Mar 02 2025

Keywords

• nonn

Comments

a(3k-1) = k.

Examples

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

Crossrefs

Cf. A006257, A054995, A225381, A321298, A378635, A380195, A008585, A381591 A381667.

Programs

Python

def UUD(n):
    return invPerm(UUDES(n))
def UUDES(n):
    l=[]
    for i in range(n):
        l.append(i+1)
    index = 0
    P=[]
    for i in range(n):
        index+=2
        index=index%len(l)
        P.append(l[index])
        l.pop(index)
    return P
def invPerm(p):
    inv = []
    for i in range(len(p)):
        inv.append(None)
    for i in range(len(p)):
        inv[p[i]-1]=i+1
    return inv
sequence = []
for i in range(1, 71):
    sequence += [str(UUD(i)[0])]
print(", ".join(sequence))

Python

def a(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 0:
        i = (i + 2)%len(J)
        q = J.pop(i)
        if q == 1: return c
        c = c+1
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Mar 24 2025