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

Author

Wolfdieter Lang, Apr 18 2013

Keywords

• nonn

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.

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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 *)

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).