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A360424 Array read by rows: row n consists of the numbers k such that k^2 + (sum of n-th powers of the digits of k^2) is a square.

%I A360424 #21 Jun 03 2024 18:37:53
%S A360424 0,0,6,0,0,89,137,6985,0,3072,0,68,8346,213202,470102,540674,1014879,
%T A360424 0,106329,0,37941,1582656,9244855,45046529,0,1239,5496,14247,490065
%N A360424 Array read by rows: row n consists of the numbers k such that k^2 + (sum of n-th powers of the digits of k^2) is a square.
%C A360424 T(n,1) = 0.
%C A360424 T(n,k) <= x where 9^n*(1+log_10(x^2)) = 2*x+1.
%e A360424 Array begins:
%e A360424   0
%e A360424   0, 6
%e A360424   0
%e A360424   0, 89, 137, 6985
%e A360424   0, 3072
%e A360424   0, 68, 8346, 213202, 470102, 540674, 1014879
%e A360424   0, 106329
%e A360424   0, 37941, 1582656, 9244855, 45046529.
%e A360424 T(4,3) = 137 is a term because 137^2 = 18769 and 18769 + 1^4 + 8^4 + 7^4 + 6^4 + 9^4 = 182^2.
%p A360424 f:= proc(d) local k, kmax;
%p A360424   kmax:= fsolve(k^2 + 9^d*(1+log[10](k^2))=(k+1)^2, k=0..infinity);
%p A360424   select(k -> issqr(k^2 + add(t^d, t=convert(k^2,base,10))), [$0..floor(kmax)])
%p A360424 end proc:
%p A360424 seq(f(d), d=0..7);
%Y A360424 Cf. A360422.
%K A360424 nonn,tabf,base,more
%O A360424 1,3
%A A360424 _Robert Israel_, Feb 06 2023