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%I #85 Dec 30 2023 03:06:32
%S 3,6,8,10,11,13,15,16,18,20,21,23,26,27,28,31,33,36,37,38,40,41,43,44,
%T 45,46,48,49,51,52,53,54,55,56,57,58,59,61,62,63,64,66,67,68,71,73,74,
%U 75,76,77,78,80,81,83,86,88,89,91,92,93
%N Numbers z such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x,y.
%C Theorem (Sierpinski, 1963): n is a term iff n^2+(n+1)^2 is a composite number. - _N. J. A. Sloane_, Feb 29 2020
%C For n > 1, A047219 is a subset of this sequence. This is because n^2 + (n+1)^2 is divisible by 5 if n is (1 or 3) mod 5 (also see A027861). - _Dmitry Kamenetsky_, Sep 02 2008
%C From _Hermann Stamm-Wilbrandt_, Sep 10 2014: (Start)
%C For n > 0, A212160 is a subset of this sequence (n^2 + (n+1)^2 is divisible by 13 if n == (2 or 10) (mod 13)).
%C For n >= 0, A212161 is a subset of this sequence (n^2 + (n+1)^2 is divisible by 17 if n == (6 or 10) (mod 17)).
%C The above are for divisibility by 5, 13, 17; notation (1,3,5), (2,10,13), (6,10,17). Divisibility by p for a and p-a-1; notation (a,p-a-1,p). These are the next tuples: (8,20,29), (15,21,37), (4,36,41), (11,41,53), ... . The corresponding sequences are a subset of this sequence (8,20,37,49,66,78,... for (8,20,29)). These sequences have no entries in the OEIS yet. For any prime of the form 4*k+1 there is exactly one of these tuples/sequences.
%C For n > 1, A000217 (triangular numbers) is a subset of this sequence (3,6,10,15,...); z=A000217(n), y=z-1, x=n.
%C For n > 0, A001652 is a subset of this sequence; z=A001652(n), x=y=A053141(n).
%C For n > 1, A001108(=A115598) is a subset of this sequence; z=A001108(n), x=A076708(n), y=x+1.
%C For n > 0, A124124(2*n+1)(=A098790(2*n)) is a subset of this sequence (6,37,218,...); z=A124124(2*n+1), x=a(n)-1, y=a(n)+1, a(m) = 6*a(m-1) - a(m-2) + 2, a(0)=0, a(1)=4.
%C (End)
%D Aviezri S. Fraenkel, Diophantine equations involving generalized triangular and tetrahedral numbers, pp. 99-114 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
%H Vincenzo Librandi, <a href="/A012132/b012132.txt">Table of n, a(n) for n = 1..1000</a>
%H H. Finner and K. Strassburger, <a href="http://dx.doi.org/10.1007/s001840100117">Increasing sample sizes do not necessarily increase the power of UMPU-tests for 2 X 2-tables</a>, Metrika, 54, 77-91, (2001).
%H Heiko Harborth, <a href="http://dx.doi.org/10.1007/978-94-015-7801-1_1">Fermat-like binomial equations</a>, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988).
%H W. Sierpinski, <a href="https://doi.org/10.14708/wm.v7i1.2383">On triangular numbers which are sums of two smaller triangular numbers</a>, (Polish), Wiadom. Mat. (2) 7 (1963), 27-28. See MR0182602.
%t Select[Range[100], !PrimeQ[#^2 + (#+1)^2]& ] (* _Jean-François Alcover_, Jan 17 2013, after _Michael Somos_ *)
%Y Complement of A027861. - _Michael Somos_, Jun 08 2000
%Y Cf. A000217, A047219, A027861, A166080.
%Y Cf. also A212160, A212161, A001652, A001108, A115598, A124124, A098790.
%K nonn
%O 1,1
%A Sander van Rijnswou (sander(AT)win.tue.nl)
%E More terms and references from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 09 2000
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