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).

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, 53, 18, 2, 15, 61, 40, 30, 68, 44, 32, 78, 29, 81, 15, 19, 76, 51, 67, 40, 19, 53, 42, 53, 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

Offset: 1

Author

Paul Weisenhorn, Jan 06 2013

Keywords

• nonn

Comments

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.

Examples

			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}.
		

References

• W. Ahrens, Das Josephusspiel, Archiv für Kulturgeschichte, Jg 11(1913), 129-151.

Links

• R. Baumann, Das Josephus-Problem, LOG IN, Heft Nr. 165, pp. 68-71, 2010 (in German).
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A187788, A206602.

Programs

Maple

stone:=proc(n1)
local n,j,k,h,z,zp: global a,m,s:
n:=2*n1: m:=m+1:
for j from 1 to n-1 do z[j]:=z[j]+1: end do:
z[n]:=1: zp:=1:
for j from 1 to n1 do
for k from 1 to (s-2) do  zp:=z[zp]: end do:
  h:=z[zp]: z[zp]:=z[z[zp]]: zp:=z[zp]:
end do:
if (h=1) then a[n1]:=1: else a[n1]:=n+2-h: end if:
end proc:
m:=0: s:=1:
while (m < 100) do
s1:=s: s:=s+1: c:=1:
for p from 2 to 100 by 2 do  p1:=p-1: p2:=p+1:
  b:=(c+s1) mod p +1:
  if (b=1) and (a[p1]=0) then stone(p1): end if:
  c:=(b+s1) mod p2 +1:
  if (c=1) and (a[p]=0) then stone(p): end if:
end do:
end do: