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

%I A292990 #16 Nov 23 2024 17:47:44
%S A292990 351,561,780,990,1176,1596,2016,2145,3321,3741,4278,4371,5565,6216,
%T A292990 6786,7503,7626,7875,8256,10296,10440,10731,11781,12561,12880,13041,
%U A292990 13695,14196,14535,14706,15576,16836,17391,17955,18915,20100,20503,20910,21321,21528
%N A292990 Numbers whose absolute difference from a triangular number is never a prime.
%C A292990 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).
%C A292990 The smallest numbers in this sequence having fewer than 8 divisors are
%C A292990 a(82) = 65341 = A000217(361) = 19^2 * 181,
%C A292990 a(248) = 354061 = A000217(841) = 29^2 * 421,
%C A292990 a(1431) = 6924781 = A000217(3721) = 61^2 * 1861,
%C A292990 a(2021) = 12708361 = A000217(5041) = 71^2 * 2521, and
%C A292990 a(2589) = 19478161 = A000217(6241) = 79^2 * 3121, each of which is a triangular number with exactly 6 divisors (A292989).
%C A292990 Conjectures:
%C A292990 (1) This sequence is a subset of the triangular numbers (A000217).
%C A292990 (2) This sequence includes no semiprimes.
%e A292990 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,
%e A292990 |k - 26| = 1 would require |k + 27|/2 = 26 or 27 (neither of which is prime),
%e A292990 |k + 27|/2 = 1 would require |k - 26| = 51 or 55 (neither of which is prime),
%e A292990 |k/2 - 13| = 1 would require |k + 27| = 51 or 55 (neither of which is prime), and
%e A292990 |k + 27| = 1 would require |k/2 - 13| = 26 or 27 (neither of which is prime),
%e A292990 so there is no triangular number T(k) such that |T(k) - 351| is prime; thus, 351 is in the sequence.
%e A292990 120 is not in the sequence because |T(13) - 120| = |91 - 120| = 29 is prime.
%Y A292990 Cf. A000040 (prime numbers), A000217 (triangular numbers).
%Y A292990 Cf. A292989 (triangular numbers having exactly 6 divisors).
%K A292990 nonn
%O A292990 1,1
%A A292990 _Jon E. Schoenfield_, Dec 08 2017