A292990 Numbers whose absolute difference from a triangular number is never a prime.

351, 561, 780, 990, 1176, 1596, 2016, 2145, 3321, 3741, 4278, 4371, 5565, 6216, 6786, 7503, 7626, 7875, 8256, 10296, 10440, 10731, 11781, 12561, 12880, 13041, 13695, 14196, 14535, 14706, 15576, 16836, 17391, 17955, 18915, 20100, 20503, 20910, 21321, 21528

Offset: 1

Author

Jon E. Schoenfield, Dec 08 2017

Keywords

• nonn

Comments

This sequence contains no primes (since any prime p has an absolute difference of p from the zeroth triangular number, A000217(0) = 0*(0+1)/2 = 0).

The smallest numbers in this sequence having fewer than 8 divisors are

a(82) = 65341 = A000217(361) = 19^2 * 181,

a(248) = 354061 = A000217(841) = 29^2 * 421,

a(1431) = 6924781 = A000217(3721) = 61^2 * 1861,

a(2021) = 12708361 = A000217(5041) = 71^2 * 2521, and

a(2589) = 19478161 = A000217(6241) = 79^2 * 3121, each of which is a triangular number with exactly 6 divisors (A292989).

Conjectures:

(1) This sequence is a subset of the triangular numbers (A000217).

(2) This sequence includes no semiprimes.

Examples

			The difference d between any triangular number T(k) = k(k+1)/2 and 351 can be factored as (k - 26) * (k + 27)/2 if k is odd, or as (k/2 - 13)*(k + 27) if k is even, so |d| cannot be prime unless |k - 26| and |k + 27|/2 are 1 and a prime, in some order, or |k/2 - 13| and |k + 27| are 1 and a prime, in some order; however,
|k - 26| = 1 would require |k + 27|/2 = 26 or 27 (neither of which is prime),
|k + 27|/2 = 1 would require |k - 26| = 51 or 55 (neither of which is prime),
|k/2 - 13| = 1 would require |k + 27| = 51 or 55 (neither of which is prime), and
|k + 27| = 1 would require |k/2 - 13| = 26 or 27 (neither of which is prime),
so there is no triangular number T(k) such that |T(k) - 351| is prime; thus, 351 is in the sequence.
120 is not in the sequence because |T(13) - 120| = |91 - 120| = 29 is prime.
		

Crossrefs

Cf. A000040 (prime numbers), A000217 (triangular numbers).

Cf. A292989 (triangular numbers having exactly 6 divisors).