This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385513 #13 Aug 08 2025 20:14:31
%S A385513 1,6,7,105,181,215,821,1907,3176,23388,55058
%N A385513 The numbers of people in the "SpellUnder-Down" variant of the Josephus problem such that the last person is freed.
%C A385513 In SpellUnder-Down dealing, we spell the number of the next card, putting a card under for each letter in the number, then we deal the next card. So we start with putting 3 cards under, for O-N-E, then deal, then 3 under for T-W-O, then deal, then 5 under for T-H-R-E-E, then deal. The dealing sequence is highly irregular because it depends on English spelling. The dealing pattern starts: UUUDUUUDUUUUUD. In the corresponding Josephus problem, we skip the next person for each under dealing, and eliminate the next person for each down dealing.
%C A385513 This sequence can be used in magic tricks with the SpellUnder-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.
%C A385513 The classical Josephus problem corresponds to under-down dealing. In this case, the last person is freed when the number of people is a power of 2 minus 1.
%C A385513 A naive probabilistic argument predicts the probability that A380204(k) = k is 1/k and expects this sequence to be infinite and distributed roughly as A002387. - _Michael S. Branicky_, Jul 24 2025
%F A385513 {k | A380204(k) = k}. - _Michael S. Branicky_, Jul 24 2025
%e A385513 Suppose there are 5 people in a circle. We start with skipping three people for O-N-E. After three people are skipped, the person number 4 is eliminated. The leftover people are 5,1,2,3 in order. Then we skip three people for T-W-O. The person number 3 eliminated, and the leftover people are 5,1,2 in order. Then we skip 5 people for T-H-R-E-E, and person number 2 is eliminated, and the leftover people are 5,1 in order. Then we skip four people for F-O-U-R. person number 5 is eliminated. Person 1 is freed. As person 1 is not last, 5 is NOT in this sequence.
%Y A385513 Cf. A005589, A006257, A182459, A225381, A321298, A378635, A380201, A380202, A380204, A380246, A380247, A380248, A385328.
%K A385513 nonn,more,word
%O A385513 1,2
%A A385513 _Tanya Khovanova_, _Nathan Sheffield_, and the MIT PRIMES STEP junior group, Jul 01 2025
%E A385513 a(10)-a(11) from _Michael S. Branicky_, Jul 24 2025