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A342154 Number of partitions of n^5 into two positive squares.

%I A342154 #17 Mar 03 2021 19:29:28
%S A342154 0,0,1,0,0,3,0,0,1,0,3,0,0,3,0,0,0,3,1,0,3,0,0,0,0,5,3,0,0,3,0,0,1,0,
%T A342154 3,0,0,3,0,0,3,3,0,0,0,3,0,0,0,0,6,0,3,3,0,0,0,0,3,0,0,3,0,0,0,18,0,0,
%U A342154 3,0,0,0,1,3,3,0,0,0,0,0,3,0,3,0,0,18,0,0,0,3,3,0,0,0,0,0,0,3,1,0,5,3,0,0,3,0
%N A342154 Number of partitions of n^5 into two positive squares.
%C A342154 a(n) > 0 if and only if n is in A000404. - _Robert Israel_, Mar 03 2021
%H A342154 Robert Israel, <a href="/A342154/b342154.txt">Table of n, a(n) for n = 0..3050</a>
%H A342154 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A342154 a(n) = A025426(A000584(n)).
%e A342154 2^5 = 32 = 4^2 + 4^2. So a(2) = 1.
%e A342154 5^5 = 3125 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2. So a(5) = 3.
%p A342154 f:= proc(n) local x,y,S;
%p A342154       S:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^5)]);
%p A342154       nops(select(t -> t[1] >= t[2] and t[2] > 0, S))
%p A342154 end proc:
%p A342154 map(f, [$0..200]); # _Robert Israel_, Mar 03 2021
%o A342154 (PARI) a(n) = my(cnt=0, m=n^5); for(k=1, sqrt(m/2), l=m-k*k; if(l>0&&issquare(l), cnt++)); cnt;
%Y A342154 Cf. A000404, A000584, A025426, A046080, A084888.
%K A342154 nonn,easy
%O A342154 0,6
%A A342154 _Seiichi Manyama_, Mar 02 2021