A338800 - OEIS

%I #23 Jul 01 2024 15:05:00

%S 16,410,6058655748,3262811042

%N Smallest number that is the sum of two distinct n-th powers of primes in two different ways.

%C The Lander, Parkin, and Selfridge conjecture implies that for n >= 5 a number can be the sum of two n-th powers of positive integers in at most one way, and in particular that a(n) does not exist for n >= 5. - _Robert Israel_, Nov 13 2020

%C a(5) > 10^31 if it exists. - _Michael S. Branicky_, Jul 01 2024

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture">Lander, Parkin, and Selfridge conjecture</a>.

%e 16 = 3 + 13 = 5 + 11.

%e 410 = 7^2 + 19^2 = 11^2 + 17^2.

%e 6058655748 = 61^3 + 1823^3 = 1049^3 + 1699^3.

%e 3262811042 = 7^4 + 239^4 = 157^4 + 227^4.

%p f:= proc(n) local S,P,p,pn,b;

%p   S:= {}:

%p   P:= {}:

%p   p:= 1:

%p   b:= infinity;

%p   do

%p    p:= nextprime(p);

%p    pn:= p^n;

%p    if pn > b then return b fi;

%p    V:= select(`<`,map(`+`,P,pn),b);

%p    newv:= V intersect S;

%p    S:= S union V;

%p    P:= P union {p^n};

%p    if newv <> {} then

%p      b:= min(newv);

%p      S:= select(`<`,S,b);

%p      P:= select(`<`,P, b);

%p    fi;

%p   od:

%p end proc:

%p map(f, [$1..4]); # _Robert Israel_, Nov 13 2020

%Y Cf. A016078, A046881, A338799.

%K nonn,hard,more

%O 1,1

%A _Ilya Gutkovskiy_, Nov 10 2020