A206602 -id:A206602 - cp's OEIS frontend

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

2, 5, 2, 4, 3, 11, 2, 3, 8, 16, 4, 21, 6, 5, 2, 11, 20, 34, 8, 15, 10, 7, 13, 11, 13, 45, 18, 23, 8, 3, 2, 25, 75, 42, 13, 5, 23, 13, 50, 16, 18, 89, 38, 8, 39, 30, 29, 38, 7, 45, 23, 137, 46, 63, 17, 48, 5, 46, 34, 140, 33, 39, 2, 28, 29, 79, 33, 48, 3, 10, 46, 120, 6, 37, 17, 8, 44, 15, 160, 20, 35, 144, 104, 179, 153, 24, 8, 265, 19, 9, 62, 7, 139, 19, 44, 93, 182, 27, 158, 185

Offset: 1

Paul Weisenhorn, Jan 06 2013

nonn

Beginning at the position A187789(n) with step a(n), (n-1) white stones were eliminated; then from the position 1 of the last white stone, n black stones were eliminated.

			n=4; WBWWBBWB; startposition=A187789(4)=8; least step=a(4)=4; elimination: white stones: {3,7,4}; black stones: {6,5,8,2}.

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

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

Cf. A187789, A206602.

First column in A321781.

s:=1: M:={}:
for n from 1 to 100 do M:=M union {n}:
while (M <> {}) do
  s1:=s: s:=s+1: f[1]:=1:
  for n from 2 to 101 do  n1:=n-1:
    f[n]:=(f[n1]+s1) mod n +1:
    if (f[n]=1) and (n1 in M) then
      a[n1]:=s: M:=M minus {n1}:
    end if:
  end do:
end do:

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).

Original entry on oeis.org

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

Paul Weisenhorn, Jan 06 2013

nonn

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.

			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}.

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

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

Cf. A187788, A206602.

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:

cp's OEIS Frontend

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

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