A000217 and related
Antreas P. Hatzipolakis
xpolakis at otenet.gr
Sat Jun 3 22:14:31 CEST 2000
D Number: A000217 (Formerly M2535 and N1002)
Sequence: 0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,
190,210,231,253,276,300,325,351,378,406,435,465,496,528,561,
595,630,666,703,741,780,820,861,903,946,990,1035,1081,1128,
1176,1225,1275
Name: Triangular numbers: C(n+1,2) = n(n+1)/2.
_________________________________________________________________________
1. Triangular numbers Tx which are the sum of two triangular numbers Ty, Tz
(with y,z <x): 6,21,36,55,66,91,120,136,....
x Tx Ty Tz
_____________________________________
3 T3 = 6 = 3 + 3 = T2 + T2
6 T6 = 21 = 6 + 15 = T3 + T5
8 T8 = 36 = 15 + 21 = T5 + T6
10 T10 = 55 = 10 + 45 = T4 + T9
11 T11 = 66 = 21 + 45 = T6 + T9
13 T13 = 91 = 36 + 55 = T8 + T10
15 T15 = 120 = 15 + 105 = T5 + T14
16 T16 = 136 = 45 + 91 = T9 + T13
............
Sequence of x's: 3,6,8,10,11,13,15,16,.... = A012132
Theorem (Sierpinski): x^2 + (x+1)^2 is composite.
x^2 (x+1)^2
-----------------------------
3^2 + 4^2 = 25 = 5 X 5
6^2 + 7^2 = 85 = 5 X 17
8^2 + 9^2 = 145 = 5 X 29
10^2 + 11^2 = 221 = 13 X 17
11^2 + 12^2 = 265 = 5 X 53
13^2 + 14^2 = 365 = 5 X 73
15^2 + 16^2 = 481 = 13 X 37
16^2 + 17^2 = 545 = 5 X 109
.......
Sierpinski's Sequence of Composites: 25,85,145,221,265,365,481,545,....
[Numbers of the form x^2 + (x+1)^2, where x's are: A012132]
2. Conjecture (Schinzel): There are infinitely many triangular numbers
which are not the sum of two triangular numbers.
Schinzel's Sequence: 1,3,10,15,28,45,78,105,153,190,.....
Antreas
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