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

%I A205791 #19 May 14 2021 16:46:52
%S A205791 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,
%T A205791 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,
%U A205791 9,7,16,58,59,60,32,7,4,24,6,66,8,68,36,70,71,4,12,74,75,16,40
%N A205791 Least positive integer k such that n divides k^5-j^5 for some j in [1,k-1].
%C A205791 For a guide to related sequences, see A204892.
%C A205791 a(n) <= n+1. If n is divisible by p^2 then a(n) <= p+n/p. - _Robert Israel_, May 14 2021
%H A205791 Robert Israel, <a href="/A205791/b205791.txt">Table of n, a(n) for n = 1..3000</a>
%e A205791 1 divides 2^5-1^5 -> k=2, j=1
%e A205791 2 divides 3^5-1^5 -> k=3, j=1
%e A205791 3 divides 4^5-1^5 -> k=4, j=1
%e A205791 4 divides 4^5-2^5 -> k=4, j=2
%e A205791 5 divides 6^5-1^5 -> k=6, j=1
%e A205791 6 divides 7^5-1^5 -> k=7, j=1
%p A205791 N:= 100: # for a(1)..a(N)
%p A205791 V:= Vector(N):
%p A205791 count:= 0:
%p A205791 for k from 1 while count < N do
%p A205791   for j from 1 to k-1 while count < N do
%p A205791     Q:= select(t -> t <= N and V[t] = 0, numtheory:-divisors(k^5-j^5));
%p A205791     if Q <> {} then
%p A205791        newcount:= nops(Q);
%p A205791        count:= count + newcount;
%p A205791        V[convert(Q,list)]:= k;
%p A205791     fi
%p A205791 od od:
%p A205791 convert(V,list); # _Robert Israel_, May 14 2021
%t A205791 s = Table[n^4, {n, 1, 120}] ;
%t A205791 lk = Table[
%t A205791   NestWhile[# + 1 &, 1,
%t A205791    Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1,
%t A205791     Length[s]}]
%t A205791 Table[NestWhile[# + 1 &, 1,
%t A205791   Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &], {j, 1, Length[lk]}]
%t A205791 (* _Peter J. C. Moses_, Jan 27 2012 *)
%t A205791 Array[(k=1;While[FreeQ[Mod[Table[k^5-j^5,{j,k-1}],#],0],k++];k)&,100] (* _Giorgos Kalogeropoulos_, May 14 2021 *)
%Y A205791 Cf. A204892, A344308.
%K A205791 nonn,look
%O A205791 1,1
%A A205791 _Clark Kimberling_, Feb 01 2012