A050798 Numbers n such that m = n^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

1, 7, 8, 12, 13, 17, 21, 22, 23, 27, 28, 30, 31, 33, 34, 37, 41, 42, 44, 46, 48, 50, 52, 53, 55, 58, 60, 62, 63, 64, 67, 75, 76, 77, 78, 80, 81, 86, 87, 88, 89, 91, 92, 96, 97, 100, 102, 103, 104, 105, 106, 108, 109, 111, 113, 114, 115, 119, 125, 127, 129, 135, 136

Offset: 1

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn
• nice

Comments

Of course m = n^2 + 1 is the sum of two squares, by definition. Here there should be just one other way to write m as a different sum of two squares.

Let p and q be primes of the form 1+4k. Then n^2+1 must be pq or 2pq. - T. D. Noe, May 27 2008

Examples

			E.g., 111^2 + 1 = 21^2 + 109^2 only.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein, MathWorld: Sum of Squares Function
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000161, A050797, A050796.

Programs

Mathematica

ok[1] = True; ok[n_] := Length[ {ToRules[ Reduce[ 1 < x <= y && n^2 + 1 == x^2 + y^2, {x, y}, Integers] ] } ] == 1; Select[ Range[136], ok] (* Jean-François Alcover, Feb 16 2012 *)

Extensions

Better definition from T. D. Noe, May 27 2008