A205791 Least positive integer k such that n divides k^5-j^5 for some j in [1,k-1].

2, 3, 4, 4, 6, 7, 8, 4, 6, 11, 3, 8, 14, 15, 16, 4, 18, 9, 20, 12, 22, 3, 24, 8, 6, 27, 6, 16, 30, 31, 2, 4, 4, 35, 36, 12, 38, 39, 40, 12, 7, 43, 44, 5, 18, 47, 48, 8, 14, 11, 52, 28, 54, 9, 7, 16, 58, 59, 60, 32, 7, 4, 24, 6, 66, 8, 68, 36, 70, 71, 4, 12, 74, 75, 16, 40

Offset: 1

Author

Clark Kimberling, Feb 01 2012

Keywords

• nonn
• look

Comments

For a guide to related sequences, see A204892.

a(n) <= n+1. If n is divisible by p^2 then a(n) <= p+n/p. - Robert Israel, May 14 2021

Examples

			1 divides 2^5-1^5 -> k=2, j=1
2 divides 3^5-1^5 -> k=3, j=1
3 divides 4^5-1^5 -> k=4, j=1
4 divides 4^5-2^5 -> k=4, j=2
5 divides 6^5-1^5 -> k=6, j=1
6 divides 7^5-1^5 -> k=7, j=1
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A204892, A344308.

Programs

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
count:= 0:
for k from 1 while count < N do
  for j from 1 to k-1 while count < N do
    Q:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(k^5-j^5));
    if Q <> {} then
       newcount:= nops(Q);
       count:= count + newcount;
       V[convert(Q,list)]:= k;
    fi
od od:
convert(V,list); # Robert Israel, May 14 2021

Mathematica

s = Table[n^4, {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 *)
Array[(k=1;While[FreeQ[Mod[Table[k^5-j^5,{j,k-1}],#],0],k++];k)&,100] (* Giorgos Kalogeropoulos, May 14 2021 *)