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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.

%I A384774 #19 Jul 30 2025 04:24:13
%S A384774 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,
%T A384774 13,21,8,30,23,20,9,16,31,17,10,31,24,35,11,34,20,27,12,28,34,49,13,
%U A384774 23,31,24,14,49,55,34,15,35,27,59,16,44,38,60,17,30,53,31
%N 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.
%C A384774 a(4k-1) = k
%C A384774 a(n) = A384770(n,1).
%e A384774 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.
%p A384774 A384774 := proc(n::integer)
%p A384774     local plist,eli,skip,ptr ;
%p A384774     plist := [seq(i,i=1..n)] ;
%p A384774     eli :=1 ;
%p A384774     skip := 3;
%p A384774     ptr := 0 ;
%p A384774     while true do
%p A384774         ptr := modp(ptr+skip,nops(plist)) ;
%p A384774         if op(ptr+1,plist) = 1 then
%p A384774             return eli ;
%p A384774         end if;
%p A384774         plist := subsop(ptr+1=NULL,plist) ;
%p A384774         eli := eli+1 ;
%p A384774     end do:
%p A384774 end proc:
%p A384774 seq(A384774(n),n=1..100) ; # _R. J. Mathar_, Jul 30 2025
%o A384774 (Python)
%o A384774 def a(n):
%o A384774     c, i, J = 1, 0, list(range(1, n+1))
%o A384774     while len(J) > 0:
%o A384774         i = (i + 3)%len(J)
%o A384774         q = J.pop(i)
%o A384774         if q == 1: return c
%o A384774         c = c+1
%o A384774 print([a(n) for n in range(1, 71)])
%Y A384774 Cf. First column of A384770.
%Y A384774 Cf. A225381, A384772, A088333.
%K A384774 nonn
%O A384774 1,2
%A A384774 _Tanya Khovanova_, _Nathan Sheffield_, and the MIT PRIMES STEP junior group, Jun 09 2025