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A260825 List of pairs of positive integers [b,c], in increasing order of b*c, whose sum and product are both squares. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 16 11:09 EDT 2021. Contains 343942 sequences. (Running on oeis4.)