A266890 Squares whose arithmetic derivative is a square.

0, 1, 4, 256, 11664, 262144, 531441, 11943936, 156250000, 544195584, 4294967296, 7119140625, 24794911296, 160000000000, 195689447424, 1129718145924, 7290000000000, 8916100448256, 10851569165584, 95367431640625, 332150625000000, 406239826673664, 494424620106921

Offset: 1

Author

Paolo P. Lava, Apr 08 2016

Keywords

• nonn

Comments

This sequence is infinite since it contains all the numbers of the form 4^(k^2). - Giovanni Resta, May 28 2016

Examples

			0' = 0 = 0^2; 1' = 0 = 0^2; 4' = 4 = 2^2; 256' = 1024 = 32^2; 11664' = 46656 = 216^2.
		

Crossrefs

Cf. A000290, A003415, A008848.

Programs

Maple

with(numtheory): P:=proc(q) local a,n,p;
for n from 0 to q do a:=n^2*add(op(2,p)/op(1,p),p=ifactors(n^2)[2]);
if trunc(sqrt(a))*trunc(sqrt(a))=a then print(n^2);  fi;
od; end: P(10^9);

Mathematica

{0, 1}~Join~Select[Range[2, 10^5]^2, IntegerQ@ Sqrt[# Total[#2/#1 & @@@ FactorInteger[#]]] &] (* Michael De Vlieger, Oct 19 2021 *)

PARI

ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k,2]/f[k,1]));
lista(nn) = {for (n=0, nn, if (issquare(ad(n^2)), print1(n^2, ", ")););} \\ Michel Marcus, Apr 08 2016

Extensions

a(17)-a(23) from Giovanni Resta, May 28 2016