A260825 List of pairs of positive integers [b,c], in increasing order of b*c, whose sum and product are both squares.

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, 144, 17, 272, 72, 72, 52, 117, 2, 3362, 45, 180, 98, 98, 81, 144, 20, 605, 64, 225, 40, 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

Offset

1,1

Comments

Pairs have been put in increasing order of b*c, thus [9, 16] comes after [2, 98] (since 9*16 < 2*98).

From Robert Israel, Mar 16 2018: (Start)

b and c are listed with b <= c.

First b*c that has more than one pair is 120^2 corresponding to [b,c] = [64, 225] and [40, 360].

For a given b*c, pairs are placed in increasing order of c. (End)

From Robert G. Wilson v, Aug 03 2015: (Start)

The squares (b*c) are 4, 64, 100, 144, 196, 324, 900, 1024, 1600, 2304, 2500, 3136, 3600, 4624, 5184, ..., .

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, ..., .

The sums (b+c) are 4, 16, 25, 25, 100, 36, 100, 64, 100, 100, 100, 400, 169, 289, 144, 169, 3364, 225, ..., . (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

[5,20] is such a pair, since 5 + 20 = 25 = 5^2 and 5*20 = 100 = 10^2.

Maple

count:= 0: Res:= NULL;
for t from 1 while count < 200 do
  tres:= NULL;
  Q:= select(q -> q^2 <= 4*t^2, numtheory:-divisors(4*t^2));
  nt:= 0:
  for q in Q do
    r:= 4*t^2/q;
    if (r-q) mod 2 <> 0 then next fi;
    s2:= (q+r)/2;
    if not issqr(s2) then next fi;
    s:= sqrt(s2);
    d:= (r-q)/2;
    b:= (s2+d)/2;
    c:= (s2-d)/2;
    count:= count+2;
    tres:= tres, [c, b];
    nt:= nt+1;
  od;
  if nt =1 then tres:= [tres]
  else  tres:= sort([tres], (a, b) -> evalb(a[2] < b[2]));
  fi;
  Res:= Res, op(map(op, tres));
od:
Res; # Robert Israel, Mar 16 2018

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A001105 (when b=c).

Sequence in context: A021441 A196066 A334574 * A330763 A138102 A187791

Adjacent sequences: A260822 A260823 A260824 * A260826 A260827 A260828

Keyword

nonn,tabf

Author

Marco Ripà, Jul 31 2015

Extensions

More terms from Robert G. Wilson v, Aug 03 2015

Status

approved