A385333 The numbers of people such that, in the variant of the Josephus problem in which three people are skipped and then one is eliminated, the last person is the last to be eliminated.

1, 21, 38, 51, 122, 163, 689, 919, 2906, 3875, 5167, 51617, 68823, 163137, 290022, 1629537, 6866858, 9155811, 16276998, 28936886, 38582515, 121939802, 162586403, 216781871, 289042495, 513853325, 685137767, 913517023, 2165373685, 12166489185, 38452113969, 121527668842

Offset

1,2

Comments

This sequence can be used in magic tricks with under-under-under-down dealing pattern. The deck sizes in this sequence guarantee that after the dealing, the last card dealt is the one that was initially on the bottom.

Links

Table of n, a(n) for n=1..32.

Yunier Bello-Cruz and Roy Quintero-Contreras, A 3-adic Recurrence for the Fixed Points of the Josephus Function J_4, arXiv:2607.01270 [math.GM], 2026. See p. 3.

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. 18.

Example

Suppose there are 5 people in a circle. After three people are skipped, the person number 4 is eliminated. The leftover people are 5,1,2,3 in order. Then person number 3 eliminated, and the leftover people are 5,1,2 in order. Then person number 5 is eliminated, and the leftover people are 1,2 in order. Then person number 2 is eliminated, and person 1 is freed. Thus, 5 is NOT in this sequence.

Crossrefs

Cf. A088333, A384770, A384772, A384774, A385327.

Cf. A000225 (for skip 1 take 1), A182459 (for skip 2 take 1).

Sequence in context: A224701 A050782 A061906 * A380980 A139768 A307278

Adjacent sequences: A385330 A385331 A385332 * A385334 A385335 A385336

Keyword

nonn,changed

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Jun 25 2025

Extensions

More terms from Jinyuan Wang, Jul 01 2025

Status

approved