A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819, 849, 883, 899, 915

Offset: 1

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn

Comments

Analogous solutions exist for the sum of two identical squares z^2-1 = 2.r^2 (e.g. 99^2-1 = 2.70^2). Values of 'z' are the terms in sequence A001541, values of 'r' are the terms in sequence A001542.

Looking at a^2 + b^2 = c^2 - 1 modulo 4, we must have a and b even and c odd. Taking a = 2u, b = 2v and c = 2w - 1 and simplifying, we get u^2 + v^2 = w(w+1). - Franklin T. Adams-Watters, May 19 2008

If n is in this sequence, then so is n^(2^k), for all k >= 0. - Altug Alkan, Apr 13 2016

Examples

			E.g. 51^2 - 1 = 10^2 + 50^2 = 22^2 + 46^2 = 34^2 + 38^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285. [Contains errors]
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A050796, A050797, A001541, A001542, A001333.

Cf. A140612, A002378.

Programs

Mathematica

t={}; Do[i=c=1; While[iJayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 13 2016 *)

PARI

select( {is_A050795(n)=#qfbsolve(Qfb(1,0,1),n^2-1,2)}, [1..999]) \\ M. F. Hasler, Mar 07 2022

Python

from itertools import islice, count
from sympy import factorint
def A050795_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        if all(map(lambda d: d[0] % 4 != 3 or d[1] % 2 == 0, factorint(k**2-1).items())):
            yield k
A050795_list = list(islice(A050795_gen(),20)) # Chai Wah Wu, Mar 07 2022

Formula

a(n) = 2*A140612(n) + 1. - Franklin T. Adams-Watters, May 19 2008

{k : A025426(k^2-1)>0}. - R. J. Mathar, Mar 07 2022