|
%I #45 Jan 04 2019 18:28:27
%S 2,2,8,8,5,20,9,16,2,98,18,18,10,90,32,32,20,80,36,64,50,50,8,392,25,
%T 144,17,272,72,72,52,117,2,3362,45,180,98,98,81,144,20,605,64,225,40,
%U 360,18,882,128,128,26,650,80,320,162,162,49,576,5,7220,144,256,200,200,37,1332,32,1568,13,4212,98,578
%N List of pairs of positive integers [b,c], in increasing order of b*c, whose sum and product are both squares.
%C Pairs have been put in increasing order of b*c, thus [9, 16] comes after [2, 98] (since 9*16 < 2*98).
%C From _Robert Israel_, Mar 16 2018: (Start)
%C b and c are listed with b <= c.
%C First b*c that has more than one pair is 120^2 corresponding to [b,c] = [64, 225] and [40, 360].
%C For a given b*c, pairs are placed in increasing order of c. (End)
%C From _Robert G. Wilson v_, Aug 03 2015: (Start)
%C The squares (b*c) are 4, 64, 100, 144, 196, 324, 900, 1024, 1600, 2304, 2500, 3136, 3600, 4624, 5184, ..., .
%C The square roots of (b*c) are 2, 8, 10, 12, 14, 18, 30, 32, 40, 48, 50, 56, 60, 68, 72, 78, 82, 90, 98, ..., .
%C The sums (b+c) are 4, 16, 25, 25, 100, 36, 100, 64, 100, 100, 100, 400, 169, 289, 144, 169, 3364, 225, ..., . (End)
%H Robert Israel, <a href="/A260825/b260825.txt">Table of n, a(n) for n = 1..10000</a>
%e [5,20] is such a pair, since 5 + 20 = 25 = 5^2 and 5*20 = 100 = 10^2.
%p count:= 0: Res:= NULL;
%p for t from 1 while count < 200 do
%p tres:= NULL;
%p Q:= select(q -> q^2 <= 4*t^2, numtheory:-divisors(4*t^2));
%p nt:= 0:
%p for q in Q do
%p r:= 4*t^2/q;
%p if (r-q) mod 2 <> 0 then next fi;
%p s2:= (q+r)/2;
%p if not issqr(s2) then next fi;
%p s:= sqrt(s2);
%p d:= (r-q)/2;
%p b:= (s2+d)/2;
%p c:= (s2-d)/2;
%p count:= count+2;
%p tres:= tres,[c,b];
%p nt:= nt+1;
%p od;
%p if nt =1 then tres:= [tres]
%p else tres:= sort([tres],(a,b) -> evalb(a[2] < b[2]));
%p fi;
%p Res:= Res, op(map(op,tres));
%p od:
%p Res; # _Robert Israel_, Mar 16 2018
%t r = Flatten[ Union@ Table[ If[ IntegerQ[ Sqrt[b + c]] && IntegerQ[ Sqrt[b*c]], {b, c}, Sequence @@ {}], {b, 10000}, {c, b, 10000}], 1]; Take[ Sort[r, #1[[1]] #1[[2]] < #2[[1]] #2[[2]] &], 36] // Flatten (* _Robert G. Wilson v_, Aug 03 2015 *)
%o (PARI) lista(nn) = {for (n=1, nn, sq = n^2; fordiv(sq, d, if ((d <= n) && issquare(d + sq/d), print1(d, ", ", sq/d, ", "));););} \\ _Michel Marcus_, Aug 03 2015
%Y Cf. A001105 (when b=c).
%K nonn,tabf
%O 1,1
%A _Marco Ripà_, Jul 31 2015
%E More terms from _Robert G. Wilson v_, Aug 03 2015
|
