A205546 -id:A205546 - cp's OEIS frontend

A205551 The least j such that n divides k^k-j^j, where k (as in A205546) is the least number for which there is such a j.

1, 1, 1, 2, 1, 2, 2, 4, 2, 4, 1, 2, 1, 2, 1, 4, 1, 2, 4, 4, 2, 1, 2, 4, 4, 1, 3, 2, 4, 4, 1, 4, 3, 4, 1, 2, 4, 1, 1, 4, 5, 2, 1, 1, 1, 2, 4, 4, 6, 4, 1, 1, 7, 3, 6, 6, 7, 4, 5, 4, 5, 2, 2, 4, 1, 3, 6, 4, 2, 6, 1, 8, 3, 1, 6, 1, 6, 3, 8, 4, 6, 5, 12, 2, 1, 4, 1, 6, 2, 6, 1, 2, 9, 4, 5, 4, 6, 6, 3

Offset: 1

Clark Kimberling, Jan 31 2012

nonn

For a guide to related sequences, see A204892.

			1 divides 2^2-1^1 -> k=2, j=1
2 divides 3^3-1^1 -> k=3, j=1
3 divides 2^2-1^1 -> k=2, j=1
4 divides 4^4-2^2 -> k=2, j=2

Cf. A204892.

s = Table[n^n, {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 *)

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

Original entry on oeis.org

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 *)

A205554 Least positive integer k such that n divides k^(k-1)-j^(j-1) for some j in [1,k-1].

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 3, 3, 4, 7, 4, 5, 5, 5, 7, 5, 5, 7, 7, 7, 4, 5, 6, 5, 6, 5, 7, 5, 12, 8, 4, 6, 5, 7, 11, 7, 9, 7, 5, 7, 8, 13, 7, 5, 7, 6, 12, 5, 8, 8, 5, 5, 7, 12, 4, 5, 7, 12, 12, 11, 12, 4, 4, 8, 7, 8, 7, 7, 7, 11, 7, 7, 9, 9, 8, 7, 5, 5, 8, 9, 9, 8, 10, 13, 7, 7, 14, 5, 5, 13, 16, 7

Offset: 1

Clark Kimberling, Feb 01 2012

nonn

For a guide to related sequences, see A204892.

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

Cf. A204892, A205546, A000169.

s = Table[n^(n-1), {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 *)

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A205551 The least j such that n divides k^k-j^j, where k (as in A205546) is the least number for which there is such a j.

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

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A205554 Least positive integer k such that n divides k^(k-1)-j^(j-1) for some j in [1,k-1].

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica