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A266890 Squares whose arithmetic derivative is a square.

%I A266890 #23 Oct 19 2021 23:50:02
%S A266890 0,1,4,256,11664,262144,531441,11943936,156250000,544195584,
%T A266890 4294967296,7119140625,24794911296,160000000000,195689447424,
%U A266890 1129718145924,7290000000000,8916100448256,10851569165584,95367431640625,332150625000000,406239826673664,494424620106921
%N A266890 Squares whose arithmetic derivative is a square.
%C A266890 This sequence is infinite since it contains all the numbers of the form 4^(k^2). - _Giovanni Resta_, May 28 2016
%e A266890 0' = 0 = 0^2; 1' = 0 = 0^2; 4' = 4 = 2^2; 256' = 1024 = 32^2; 11664' = 46656 = 216^2.
%p A266890 with(numtheory): P:=proc(q) local a,n,p;
%p A266890 for n from 0 to q do a:=n^2*add(op(2,p)/op(1,p),p=ifactors(n^2)[2]);
%p A266890 if trunc(sqrt(a))*trunc(sqrt(a))=a then print(n^2);  fi;
%p A266890 od; end: P(10^9);
%t A266890 {0, 1}~Join~Select[Range[2, 10^5]^2, IntegerQ@ Sqrt[# Total[#2/#1 & @@@ FactorInteger[#]]] &] (* _Michael De Vlieger_, Oct 19 2021 *)
%o A266890 (PARI) ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k,2]/f[k,1]));
%o A266890 lista(nn) = {for (n=0, nn, if (issquare(ad(n^2)), print1(n^2, ", ")););} \\ _Michel Marcus_, Apr 08 2016
%Y A266890 Cf. A000290, A003415, A008848.
%K A266890 nonn
%O A266890 1,3
%A A266890 _Paolo P. Lava_, Apr 08 2016
%E A266890 a(17)-a(23) from _Giovanni Resta_, May 28 2016