

A012132


Numbers z such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x,y.


9



3, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 26, 27, 28, 31, 33, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 71, 73, 74, 75, 76, 77, 78, 80, 81, 83, 86, 88, 89, 91, 92, 93
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OFFSET

1,1


COMMENTS

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
From Hermann StammWilbrandt, Sep 10 2014: (Start)
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)).
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)).
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 pa1; notation (a,pa1,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.
For n > 1, A000217 (triangular numbers) is a subset of this sequence (3,6,10,15,...); z=A000217(n), y=z1, x=n.
For n > 0, A001652 is a subset of this sequence; z=A001652(n), x=y=A053141(n).
For n > 1, A001108(=A115598) is a subset of this sequence; z=A001108(n), x=A076708(n), y=x+1.
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(m1)  a(m2) + 2, a(0)=0, a(1)=4.
(End)


REFERENCES

Aviezri S. Fraenkel, Diophantine equations involving generalized triangular and tetrahedral numbers, pp. 99114 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Finner and K. Strassburger, Increasing sample sizes do not necessarily increase the power of UMPUtests for 2 X 2tables, Metrika, 54, 7791, (2001).
Heiko Harborth, Fermatlike binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 15 (1988).


MATHEMATICA

Select[Range[100], !PrimeQ[#^2 + (#+1)^2]& ] (* JeanFrançois Alcover, Jan 17 2013, after Michael Somos *)


CROSSREFS

Complement of A027861.  Michael Somos, Jun 08 2000
Cf. A047219, A027861.
Cf. A212160, A212161.
Cf. A001652, A001108, A115598, A124124, A098790.
Cf. A000217.
Sequence in context: A287362 A055073 A328505 * A108769 A286754 A186387
Adjacent sequences: A012129 A012130 A012131 * A012133 A012134 A012135


KEYWORD

nonn


AUTHOR

Sander van Rijnswou (sander(AT)win.tue.nl)


EXTENSIONS

More terms and references from Klaus Strassburger (strass(AT)ddfi.uniduesseldorf.de), Feb 09 2000


STATUS

approved



