A224770 Numbers that are the primitive sum of two squares in exactly two ways.

65, 85, 130, 145, 170, 185, 205, 221, 265, 290, 305, 325, 365, 370, 377, 410, 425, 442, 445, 481, 485, 493, 505, 530, 533, 545, 565, 610, 629, 650, 685, 689, 697, 725, 730, 745, 754, 785, 793, 845, 850, 865, 890, 901, 905, 925, 949, 962, 965, 970

Offset

1,1

Comments

These are the increasingly ordered numbers a(n) which satisfy A193138(a(n)) = 2.

Neither the order of the squares nor the signs of the numbers to be squared are taken into account. The two squares are necessarily distinct and each is nonzero.

This sequence is a proper subsequence of A000404.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

a(n) = a^2 + b^2, a and integers, 0 < a < b and gcd(a,b) = 1 in exactly two ways. These representations of a(n) are denoted by two different pairs (a,b).

Example

n=1,   65:  (1, 8),  (4, 7),
n=2,   85:  (2, 9),  (6, 7),
n=3,  130:  (3, 11), (7, 9),
n=4,  145:  (1, 12), (8, 9),
n=5,  170:  (1, 13), (7, 11),
n=6,  185:  (4, 13), (8, 11),
n=7,  205:  (3, 14), (6, 13),
n=8,  221:  (5, 14), (10, 11),
n=9,  265:  (3, 16), (11, 12),
n=10, 290:  (1, 17), (11, 13).

Mathematica

nn = 35; t = Sort[Select[Flatten[Table[If[GCD[a, b] == 1, a^2 + b^2, 0], {a, nn}, {b, a, nn}]], 0 < # <= nn^2 &]]; Transpose[Select[Tally[t], #[[2]] == 2 &]][[1]] (* T. D. Noe, Apr 20 2013 *)

Crossrefs

Cf. A224450 (one way), A193138 (multiplicities), A000404, A024509.

Sequence in context: A071011 A165158 A084648 * A274044 A024409 A131574

Adjacent sequences: A224767 A224768 A224769 * A224771 A224772 A224773

Keyword

nonn

Author

Wolfdieter Lang, Apr 18 2013

Status

approved