A119665 Sign in the term (2q +/- 1) for triangular numbers of the form q * (2q +/- 1) where both factors are primes (or prime powers).

-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, 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, -1, -1, 1, -1, 1, -1, -1

Offset

1

Comments

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

Links

Table of n, a(n) for n=1..103.

Example

a(1) = -1 since  1 = 1*1 and 1 = 2*1 - 1, a(2) = +1 since  3 = 1*3 and 3 = 2*1 + 1,
a(3) = -1 since  6 = 2*3 and 3 = 2*2 - 1, a(4) = +1 since 10 = 2*5 and 5 = 2*2 + 1,
a(5) = -1 since 15 = 3*5 and 5 = 2*3 - 1, a(6) = +1 since 21 = 3*7 and 7 = 2*3 + 1,
a(7) = -1 since 28 = 4*7 and 7 = 2*4 - 1, a(8) = +1 since 36 = 4*9 and 9 = 2*4 + 1,
a(9) = -1 since 45 = 5*9 and 9 = 2*5 - 1, a(10) = 1 since 55 = 5*11 and 11 = 2*5 + 1;
66 and 78 are the first triangular numbers not equal to a product of prime powers q*(2q+-1);
a(11) = -1 since 91 = 7*13 and 13 = 2*7 - 1;
105 and 120 aren't of the required form, either;
a(12) = +1 since 136 = 8*17 and 17 = 2*8 + 1,
a(13) = -1 since 153 = 9*17 and 17 = 2*9 - 1,
a(14) = +1 since 171 = 9*19 and 19 = 2*9 + 1;
now 190, 210 and 231 aren't of the required form, which yields the first a(n) = a(n-1):
a(15) = +1 since 253 = 11*23 and 23 = 2*11 + 1. - M. F. Hasler, Apr 21 2015

Prog

(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

Crossrefs

Cf. A068443, A000217.

Sequence in context: A143431 A064179 A065357 * A121241 A122188 A158388

Adjacent sequences: A119662 A119663 A119664 * A119666 A119667 A119668

Keyword

sign

Author

Greg Huber, Jul 28 2006

Extensions

Missing and wrong terms and offset corrected; more terms added by M. F. Hasler, Apr 21 2015

Status

approved