A057898 Largest number such that n = m^a(n) - a(n) with m a positive integer; i.e., where (n + a(n))^(1/a(n)) is a positive integer.

1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Henry Bottomley, Sep 26 2000

Keywords

• nonn

Comments

It may be that positive integers can be written as n = m^k - k (with m and k > 1) in at most one way [checked up to 10000] as well as with k = 1 and m = n+1.

Examples

			a(5) = 3 since 5 = 2^3 - 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A057897, A057899.

Programs

Maple

N:= 200: # for a(1)..a(N)
V:= Vector(N,1):
for k from 2 while 2^k-k <= N do
  for m from 2 do
    v:= m^k-k;
    if v > N then break fi;
    V[v]:= k;
  od;
od:
convert(V,list); # Robert Israel, Sep 04 2020