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

1,2

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

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

Example

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

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 *)

Crossrefs

Cf. A000161, A050797, A050796.

Sequence in context: A105740 A050796 A106630 * A260628 A280999 A024706

Adjacent sequences: A050795 A050796 A050797 * A050799 A050800 A050801

Keyword

nonn,nice

Author

Patrick De Geest, Sep 15 1999

Extensions

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

Status

approved