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

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66 and 78 are the first triangular numbers not of the form "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 this form, either;

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

(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

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;

-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

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Here, 1 is considered as a prime power.

a(61) = -1 since 1 = 1*1 and 1 = 2*1 - 1, a(2) = +1 since 21 = 3 = 1*7 3 and 7 3 = 2*3 1 + 1.,

a(73) = -1 since 28 6 = 2*3 and 3 = 2*2 - 1, a(4) = +1 since 10 = 2*7 5 and 7 5 = 2*4 - 2 + 1.,

a(85) = -1 since 15 = 3*5 and 5 = 2*3 - 1, a(6) = +1 since 36 21 = 43*9 7 and 9 7 = 2*4 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 of the form "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 this 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

(PARI) for(q=1, 999, isprimepower(q)&&forstep(j=-1, 1, 2, isprimepower(q*2+j)+q*2+j==1&&print1(j", ")) \\ M. F. Hasler, Apr 21 2015

Cf. A068443, A000217.

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

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_Greg Huber (huber(AT)alum.mit.edu), _, Jul 28 2006