A205561 -id:A205561 - cp's OEIS frontend

A378188 Record values in A205561.

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;

A378189 Positions of records in A205561.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 37, 83, 137, 173, 193, 269, 311, 479, 607, 673, 1019, 1427, 1523, 1613, 3391, 3527, 4817, 5021, 5623, 9887, 14891, 15823, 21701, 22727, 24439, 26399, 27581, 28771, 29339, 35491, 37967, 49207, 51157, 52639, 54799, 64303, 93077, 104323, 115279, 116981, 117881, 135209, 157177

Offset: 1

Robert Israel, Nov 19 2024

nonn

Numbers m such that there is k such that for every m' < m, there exist j and k' such that 1 <= j < k' <= k and m' divides (2*k')! - (2*j)!, but there do not exist j and k' such that 1 <= j < k' <= k and m divides (2*k')! - (2*j)!.

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

Cf. A205561, A378188.

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:
J:= 1: m:= 2: count:= 1:
for k from 2 while count < 70 do
  v:= f(k);
  if v > m then J:= J, k; count:= count+1; m:= v;
  fi
od:
J;

A205562 Least positive integer j such that n divides (2k)!-(2j)!, where k, as in A205561, is the least number for which there is such a j.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 3, 3, 1, 2, 7, 4, 3, 3, 9, 3, 1, 3, 4, 1, 2, 2, 3, 7, 5, 4, 2, 3, 1, 4, 3, 9, 4, 3, 19, 1, 7, 3, 4, 4, 1, 3, 3, 2, 2, 3, 4, 3, 9, 7, 1, 5, 3, 4, 5, 2, 12, 3, 4, 1, 4, 4, 7, 3, 1, 9, 2, 4, 2, 3, 2, 19, 3, 5, 6, 7, 2, 3, 5, 4, 12, 4, 9, 1, 2, 3, 4, 3, 7, 2, 6, 2, 5, 4, 1

Offset: 1

Clark Kimberling, Feb 01 2012

nonn

For a guide to related sequences, see A204892.

			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

Cf. A204892, A205551.

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 *)

cp's OEIS Frontend

A378188 Record values in A205561.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A378189 Positions of records in A205561.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A205562 Least positive integer j such that n divides (2k)!-(2j)!, where k, as in A205561, is the least number for which there is such a j.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica