A240939 - OEIS

%I #26 Aug 01 2024 18:07:14

%S 0,2,2,1,1,9,1,81,729,225,324,39169,82944,176400,215296,3444736,

%T 26167684,114349225,255004929,1158920361,11638526761,42128246889,

%U 191052974116,97216010329,2430400258225,1553580508516,4666092737476,565986718738441,2137864362693921,5112360635841936

%N Least number k >= 0 such that n! + k is a perfect power.

%C The only n <= 805 where n! + a(n) is not a square is 3. - _Robert Israel_, Aug 01 2024

%H Robert Israel, <a href="/A240939/b240939.txt">Table of n, a(n) for n = 1..805</a>

%p f:= proc(n) local v,m,p,r;

%p    m:= infinity;

%p    v:= n!;

%p    p:= 1;

%p    do

%p      p:= nextprime(p);

%p      if 2^p >= m+v then break fi;

%p      r:= ceil(v^(1/p))^p - v;

%p      if r < m then m:= r fi;

%p    od;

%p    m

%p end proc:

%p map(f, [$1..50]);

%t nextPerfectPower[n_] := Min@ Table[(Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]; f[n_] := nextPerfectPower[n!] - n!; f[1] = 0; Array[f, 30] (* _Robert G. Wilson v_, Aug 04 2014 *)

%o (PARI)

%o a(n)=for(k=0,10^10,s=n!+k;if(ispower(s)||s==1,return(k)))

%o n=1;while(n<50,print1(a(n),", ");n++)

%o (PARI)

%o a(n)=for(k=1,n!,if(2^k>n!,kk=k;break));if(kk==1,return(0));L=List([]);for(i=2,kk,listinsert(L,ceil(n!^(1/i))^i-n!,1));listsort(L);L[1]

%o vector(40, n, a(n)) \\ faster program

%Y Cf. A068869, A240937.

%K nonn

%O 1,2

%A _Derek Orr_, Aug 03 2014

%E a(18) onward from _Robert G. Wilson v_, Aug 04 2014