OFFSET
1,1
COMMENTS
The first integer using a power with a composite base is 40.
The first integer where all powers have a composite base is 360.
The first integer represented in three ways is 153.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
40 = 2^3+2^5 = 2^2+6^2.
360 = 6^2+18^2 = 6^3+12^2.
153 = 2^5+11^2 = 2^7+5^2 = 3^2+12^2.
468 = 3^3 + 21^2 = 5^3 + 7^3 = 12^2 + 18^2 = 15^2 + 3^5.
1625 = 2^8 + 37^2 = 5^2 + 40^2 = 5^4 + 10^3 = 20^2 + 35^2 = 28^2 + 29^2.
MAPLE
N:= 400: # for terms <= N
Pow:= sort(convert({seq(seq(a^b, b=2..ilog[a](N)), a=2..isqrt(N))}, list)):
V:= Vector(N):
for i from 1 to nops(Pow) do
for j from 1 to i-1 do
r:= Pow[i]+Pow[j];
if r <= N then V[r]:= V[r]+1 fi
od od:
select(t -> V[t] >= 2, [$1..N]); # Robert Israel, Oct 16 2025
PROG
(PARI) b(n)=sum(k=2, sqrtint(n\2), if(!ispower(k), sum(e=2, logint((n-1)\2, k), ispower(n-k^e)!=0 )))
select(k->b(k)>=2, [1..500]) \\ Andrew Howroyd, Oct 08 2025
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Duckett, Oct 06 2025
STATUS
approved
