

A012132


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


10



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

Theorem (Sierpinski, 1963): n is a term iff n^2+(n+1)^2 is a composite number.  N. J. A. Sloane, Feb 29 2020
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
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, 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.
Sierpinski, W., On triangular numbers which are sums of two smaller triangular numbers (Polish), Wiadom. Mat. (2) 7 (1963), 2728. See MR0182602. [From Victor S. Miller]


LINKS



MATHEMATICA



CROSSREFS



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



