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A266890
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Squares whose arithmetic derivative is a square.
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2
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0, 1, 4, 256, 11664, 262144, 531441, 11943936, 156250000, 544195584, 4294967296, 7119140625, 24794911296, 160000000000, 195689447424, 1129718145924, 7290000000000, 8916100448256, 10851569165584, 95367431640625, 332150625000000, 406239826673664, 494424620106921
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OFFSET
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1,3
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COMMENTS
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This sequence is infinite since it contains all the numbers of the form 4^(k^2). - Giovanni Resta, May 28 2016
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LINKS
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EXAMPLE
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0' = 0 = 0^2; 1' = 0 = 0^2; 4' = 4 = 2^2; 256' = 1024 = 32^2; 11664' = 46656 = 216^2.
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MAPLE
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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);
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MATHEMATICA
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{0, 1}~Join~Select[Range[2, 10^5]^2, IntegerQ@ Sqrt[# Total[#2/#1 & @@@ FactorInteger[#]]] &] (* Michael De Vlieger, Oct 19 2021 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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