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.
-1, 65, 1009, 6834, 1158224, 19198660, 1518471174, 301963223843, 14599274102522, 1601155487573222
Offset: 1
Examples
a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2. a(3) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.
Programs
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Maple
f:= proc(n) uses priqueue; local pq,w,t,g,i,count,newt; g:= proc(t) local i; [-add((t[i]+i)^n,i=1..n),op(t)] end proc; w:= [0$(n+1)]; initialize(pq); insert(g([0$n]),pq); do t:= extract(pq); if t[1] = w[1] then return -t[1] fi; w:= t; for i from 2 to n+1 do if t[i]=t[-1] then newt:= g(t[2..-1] + [0$(i-2),1$(n+2-i)]); insert(newt,pq); fi od od; end proc: -1, seq(f(n),n=2..10); # Robert Israel, Jul 01 2024
Extensions
a(9)-a(10) from Robert Israel, Jul 01 2024