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A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way. 9
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 (list; graph; refs; listen; history; text; internal format)
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

Index entries for sequences related to sums of squares

FORMULA

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

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 *)

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

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Last modified July 21 17:20 EDT 2019. Contains 325198 sequences. (Running on oeis4.)