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A187789 a(n) is the start position for a Sankt-Petrus-game with n white and n black stones and the least step A187788(n).

%I A187789 #23 Jan 19 2023 09:35:13
%S A187789 2,3,2,8,5,5,2,7,1,17,15,4,8,1,2,30,26,11,35,7,26,27,23,44,24,30,6,39,
%T A187789 53,18,2,15,61,40,30,68,44,32,78,29,81,15,19,76,51,67,40,19,53,42,53,
%U A187789 3,74,103,73,35,105,78,110,105,76,61,2,5,48,128,82,36,37,63,88,87,31,123,93,126,2,1,156,89,33,160,90,135,124,136,145,79,42,26,104,94,67,44,186,30,133,137,40,118
%N A187789 a(n) is the start position for a Sankt-Petrus-game with n white and n black stones and the least step A187788(n).
%C A187789 Beginning at the position a(n) with the least step A187788(n) (n-1) white stones were eliminated; then starting at the position 1 of the last white stone, n black stones were eliminated.
%D A187789 W. Ahrens, Das Josephusspiel, Archiv für Kulturgeschichte, Jg 11(1913), 129-151.
%H A187789 R. Baumann, <a href="https://www.yumpu.com/de/document/read/460296/nr-165">Das Josephus-Problem</a>, LOG IN, Heft Nr. 165, pp. 68-71, 2010 (in German).
%H A187789 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%e A187789 n=8; WWBBWBBWWBBWWBWB; step=A187788(8)=3; start=a(8)=7; elimination: white stones: {9,12,15,2,5,8,13}, black stones: {4,10,16,6,14,7,3,11}.
%p A187789 stone:=proc(n1)
%p A187789 local n,j,k,h,z,zp: global a,m,s:
%p A187789 n:=2*n1: m:=m+1:
%p A187789 for j from 1 to n-1 do z[j]:=z[j]+1: end do:
%p A187789 z[n]:=1: zp:=1:
%p A187789 for j from 1 to n1 do
%p A187789 for k from 1 to (s-2) do  zp:=z[zp]: end do:
%p A187789   h:=z[zp]: z[zp]:=z[z[zp]]: zp:=z[zp]:
%p A187789 end do:
%p A187789 if (h=1) then a[n1]:=1: else a[n1]:=n+2-h: end if:
%p A187789 end proc:
%p A187789 m:=0: s:=1:
%p A187789 while (m < 100) do
%p A187789 s1:=s: s:=s+1: c:=1:
%p A187789 for p from 2 to 100 by 2 do  p1:=p-1: p2:=p+1:
%p A187789   b:=(c+s1) mod p +1:
%p A187789   if (b=1) and (a[p1]=0) then stone(p1): end if:
%p A187789   c:=(b+s1) mod p2 +1:
%p A187789   if (c=1) and (a[p]=0) then stone(p): end if:
%p A187789 end do:
%p A187789 end do:
%Y A187789 Cf. A187788, A206602.
%K A187789 nonn
%O A187789 1,1
%A A187789 _Paul Weisenhorn_, Jan 06 2013