OFFSET
1,1
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.
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Dec 08 2017
STATUS
approved