A386312 -id:A386312 - cp's OEIS frontend

A386305 Numbers of people such that the first person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

1, 2, 3, 18, 22, 171, 195, 234, 1262, 2136, 6040, 42545, 353067, 1332099, 1447753, 2789475, 3635021, 7857445, 9224024, 17128159, 27666710, 29279638

Offset: 1

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Aug 20 2025

This sequence can also be described in terms of "AP dealing", in which one deals a deck of N cards into a new deck by moving one card to the bottom, dealing out the next card on top of the new deck, moving two cards to the bottom, etc. This sequence consists of all the deck sizes such that the top card remains the same after AP dealing.

Numbers k such that A291317(k) = 1.

			Suppose there are 5 people in a circle. We start with skipping one person and eliminating the next (person number 2). The leftover people are 3,4,5,1 in order. Then we skip two people and eliminate person number 5. The leftover people are 1,3,4 in order. Then we skip three people and person number 1 is eliminated. The leftover people are 3,4 in order. Then we skip four people and eliminate person number 3. Person 4 is freed. As person 1 is not freed, 5 is NOT in this sequence.

Cf. A081614, A291317, A385328, A386312, A386643.

def F(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 1:
        i = (i + c) % len(J)
        q = J.pop(i)
        c = c + 1
    return J[0]
print([n for n in range(1, 100000) if F(n) == 1])

a(20)-a(22) from Jinyuan Wang, Aug 31 2025

cp's OEIS Frontend

A386305 Numbers of people such that the first person is freed in the variant of the Josephus problem in which one person is skipped, then one is eliminated, then two people are skipped and one eliminated, then three people are skipped and so on.

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