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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. 10
 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 E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285. [Contains errors] 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 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

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Last modified July 3 18:10 EDT 2022. Contains 355055 sequences. (Running on oeis4.)