A260825 - OEIS

%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