A378189 -id:A378189 - cp's OEIS frontend

A205561 Least positive integer k such that n divides (2k)! - (2j)! for some j in [1,k-1].

2, 2, 3, 3, 4, 3, 5, 3, 4, 4, 2, 3, 8, 5, 4, 4, 10, 4, 4, 4, 5, 2, 4, 3, 4, 8, 6, 5, 3, 4, 5, 5, 4, 10, 5, 4, 20, 4, 8, 4, 11, 5, 10, 4, 4, 4, 5, 4, 6, 4, 10, 8, 6, 6, 4, 5, 8, 3, 13, 4, 8, 5, 5, 5, 8, 4, 16, 10, 4, 5, 7, 4, 4, 20, 4, 8, 7, 8, 11, 4, 6, 11, 22, 5, 10, 10, 3, 4, 5, 4, 8, 4

Offset: 1

Clark Kimberling, Feb 01 2012

nonn

For a guide to related sequences, see A204892.
From Robert Israel, Nov 20 2024: (Start)
a(n) <= ceil(A002034(n)/2) + 1.
The last occurrence of k >= 2 in the sequence is a((2*k)! - 2) = k. (End)

			1 divides (2*2)!-(2*1)! -> k=2, j=1
2 divides (2*2)!-(2*1)! -> k=2, j=1
3 divides (2*3)!-(2*2)! -> k=3, j=2
4 divides (2*3)!-(2*2)! -> k=3, j=2
5 divides (2*4)!-(2*3)! -> k=4, j=3

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002034, A010050, A204892, A205546, A378188, A378189.

f:= proc(n) local S,j,x;
  S:= {}:
  x:= 1:
  for j from 1 do
    x:=x*2*j*(2*j-1) mod n;
    if member(x,S) then return j fi;
    S:= S union {x}
  od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 18 2024

s = Table[(2n)!, {n, 1, 120}];
lk = Table[NestWhile[# + 1 &, 1,
   Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}]
Table[NestWhile[# + 1 &, 1,
  Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &],
{j, 1, Length[lk]}]
(* Peter J. C. Moses, Jan 27 2012 *)

A378188 Record values in A205561.

Original entry on oeis.org

2, 3, 4, 5, 8, 10, 20, 22, 24, 29, 34, 36, 49, 59, 72, 76, 90, 108, 110, 144, 162, 173, 175, 189, 281, 410, 413, 473, 478, 511, 512, 513, 539, 555, 632, 639, 783, 790, 794, 820, 944, 1096, 1153, 1178, 1226, 1264, 1413, 1438, 1622, 1633, 1689, 1717, 1801, 1892, 1982, 2002, 2057, 2446, 2521, 2592

Offset: 1

Robert Israel, Nov 19 2024

nonn

Numbers m such that there exist j and k such that 1 <= j < m and (2*m)! - (2*j)! is divisible by k, but for all m' < m there is no j' with 1 <= j' < m' and (2*m')! - (2*j')! divisible by k, and for all k' with 1 <= k' < k there exist j' and m' with 1 <= j' < m' < m and (2*m')! - (2*j')! divisible by k'.

			a(5) = 8 is a term because A205561(13) = 8, but A205561(n) < 8 for all n < 13.

Robert Israel, Table of n, a(n) for n = 1..113

Cf. A205561, A378189.

f:= proc(n) local S,j,x;
  S:= {}:
  x:= 1:
  for j from 1 do
    x:=x*2*j*(2*j-1) mod n;
    if member(x,S) then return j fi;
    S:= S union {x}
  od
end proc:
R:= 2: m:= 2: count:= 1:
for k from 2 while count < 70 do
  v:= f(k);
  if v > m then R:= R,v; count:= count+1; m:= v;
  fi
od:
R;

cp's OEIS Frontend

A205561 Least positive integer k such that n divides (2k)! - (2j)! for some j in [1,k-1].

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