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A374256 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.

%I A374256 #14 Jul 02 2024 02:12:56
%S A374256 -1,65,1009,6834,1158224,19198660,1518471174,301963223843,
%T A374256 14599274102522,1601155487573222
%N A374256 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 2 ways, or -1 if no such number exists.
%e A374256 a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
%e A374256 a(3) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.
%p A374256 f:= proc(n) uses priqueue;
%p A374256   local pq,w,t,g,i,count,newt;
%p A374256   g:= proc(t) local i; [-add((t[i]+i)^n,i=1..n),op(t)] end proc;
%p A374256   w:= [0$(n+1)];
%p A374256   initialize(pq);
%p A374256   insert(g([0$n]),pq);
%p A374256   do
%p A374256     t:= extract(pq);
%p A374256     if t[1] = w[1] then return -t[1] fi;
%p A374256     w:= t;
%p A374256     for i from 2 to n+1 do
%p A374256         if t[i]=t[-1] then
%p A374256           newt:= g(t[2..-1] + [0$(i-2),1$(n+2-i)]);
%p A374256         insert(newt,pq);
%p A374256   fi od od;
%p A374256 end proc:
%p A374256 -1, seq(f(n),n=2..10); # _Robert Israel_, Jul 01 2024
%Y A374256 Cf. A025303, A025400, A091414, A146756, A374226, A374257.
%K A374256 sign,more
%O A374256 1,2
%A A374256 _Ilya Gutkovskiy_, Jul 01 2024
%E A374256 a(9)-a(10) from _Robert Israel_, Jul 01 2024