A240939 Least number k >= 0 such that n! + k is a perfect power.

0, 2, 2, 1, 1, 9, 1, 81, 729, 225, 324, 39169, 82944, 176400, 215296, 3444736, 26167684, 114349225, 255004929, 1158920361, 11638526761, 42128246889, 191052974116, 97216010329, 2430400258225, 1553580508516, 4666092737476, 565986718738441, 2137864362693921, 5112360635841936

Offset: 1

Author

Derek Orr, Aug 03 2014

Keywords

• nonn

Comments

The only n <= 805 where n! + a(n) is not a square is 3. - Robert Israel, Aug 01 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..805

Crossrefs

Cf. A068869, A240937.

Programs

Maple

f:= proc(n) local v,m,p,r;
   m:= infinity;
   v:= n!;
   p:= 1;
   do
     p:= nextprime(p);
     if 2^p >= m+v then break fi;
     r:= ceil(v^(1/p))^p - v;
     if r < m then m:= r fi;
   od;
   m
end proc:
map(f, [$1..50]);

Mathematica

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 *)

PARI

a(n)=for(k=0,10^10,s=n!+k;if(ispower(s)||s==1,return(k)))
n=1;while(n<50,print1(a(n),", ");n++)

PARI

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]
vector(40, n, a(n)) \\ faster program

Extensions

a(18) onward from Robert G. Wilson v, Aug 04 2014