%I #11 Apr 21 2015 22:48:09

%S -1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,1,-1,1,-1,-1,1,-1,-1,1,-1,-1,-1,1,

%T -1,1,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,1,-1,-1,-1,1,1,-1,1,-1,-1,-1,1,1,1,

%U 1,-1,1,-1,1,-1,-1,-1,-1,1,-1,-1,1,-1,1,-1,1,1,-1,1,-1,-1,1,-1,-1,-1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,1,-1,-1,-1,-1,1,-1,1,-1,-1

%N Sign in the term (2q +/- 1) for triangular numbers of the form q * (2q +/- 1) where both factors are primes (or prime powers).

%C As always, 1 is considered to be a prime power.

%e a(1) = -1 since 1 = 1*1 and 1 = 2*1 - 1, a(2) = +1 since 3 = 1*3 and 3 = 2*1 + 1,

%e a(3) = -1 since 6 = 2*3 and 3 = 2*2 - 1, a(4) = +1 since 10 = 2*5 and 5 = 2*2 + 1,

%e a(5) = -1 since 15 = 3*5 and 5 = 2*3 - 1, a(6) = +1 since 21 = 3*7 and 7 = 2*3 + 1,

%e a(7) = -1 since 28 = 4*7 and 7 = 2*4 - 1, a(8) = +1 since 36 = 4*9 and 9 = 2*4 + 1,

%e a(9) = -1 since 45 = 5*9 and 9 = 2*5 - 1, a(10) = 1 since 55 = 5*11 and 11 = 2*5 + 1;

%e 66 and 78 are the first triangular numbers not equal to a product of prime powers q*(2q+-1);

%e a(11) = -1 since 91 = 7*13 and 13 = 2*7 - 1;

%e 105 and 120 aren't of the required form, either;

%e a(12) = +1 since 136 = 8*17 and 17 = 2*8 + 1,

%e a(13) = -1 since 153 = 9*17 and 17 = 2*9 - 1,

%e a(14) = +1 since 171 = 9*19 and 19 = 2*9 + 1;

%e now 190, 210 and 231 aren't of the required form, which yields the first a(n) = a(n-1):

%e a(15) = +1 since 253 = 11*23 and 23 = 2*11 + 1. - _M. F. Hasler_, Apr 21 2015

%o (PARI) for(q=1,999,(isprimepower(q)||q==1)&&forstep(j=-1,1,2,(isprimepower(q*2+j)||q*2+j==1)&&print1(j","))) \\ _M. F. Hasler_, Apr 21 2015

%Y Cf. A068443, A000217.

%K sign

%O 1

%A _Greg Huber_, Jul 28 2006

%E Missing and wrong terms and offset corrected; more terms added by _M. F. Hasler_, Apr 21 2015