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

1,1

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

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

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

Example

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

Mathematica

t={}; Do[i=c=1; While[i<n&&c!=0, If[IntegerQ[Sqrt[n^2-1-i^2]], c=0; AppendTo[t, n]]; i++], {n, 3, 920}]; t (* Jayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Apr 13 2016 *)

Prog

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

Crossrefs

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

Cf. A140612, A002378.

Sequence in context: A240094 A174180 A106676 * A050797 A103967 A032400

Adjacent sequences: A050792 A050793 A050794 * A050796 A050797 A050798

Keyword

nonn

Author

Patrick De Geest, Sep 15 1999

Status

approved