A292990 - OEIS

%I #16 Nov 23 2024 17:47:44

%S 351,561,780,990,1176,1596,2016,2145,3321,3741,4278,4371,5565,6216,

%T 6786,7503,7626,7875,8256,10296,10440,10731,11781,12561,12880,13041,

%U 13695,14196,14535,14706,15576,16836,17391,17955,18915,20100,20503,20910,21321,21528

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

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

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

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

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

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

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

%C Conjectures:

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

%C (2) This sequence includes no semiprimes.

%e 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 |k - 26| = 1 would require |k + 27|/2 = 26 or 27 (neither of which is prime),

%e |k + 27|/2 = 1 would require |k - 26| = 51 or 55 (neither of which is prime),

%e |k/2 - 13| = 1 would require |k + 27| = 51 or 55 (neither of which is prime), and

%e |k + 27| = 1 would require |k/2 - 13| = 26 or 27 (neither of which is prime),

%e so there is no triangular number T(k) such that |T(k) - 351| is prime; thus, 351 is in the sequence.

%e 120 is not in the sequence because |T(13) - 120| = |91 - 120| = 29 is prime.

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

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

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Dec 08 2017