A330809 Triangular numbers having exactly 8 divisors.

66, 78, 105, 136, 190, 231, 351, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1431, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051

Offset: 1

Author

Jon E. Schoenfield, Jan 11 2020

Keywords

• nonn

Comments

Terms may be categorized as belonging to the following types:

type 1: products of 3 distinct primes p,q,r such that 2*p*q + 1 = r: 78, 406, 465, ... (27108 of the first 100000 terms);

type 2: products of 3 distinct primes p,q,r such that 2*p*q - 1 = r: 66, 190, 435, ... (26848 of the first 100000 terms);

type 3: products of 3 distinct primes p,q,r such that p*q + 1 = 2*r: 231, 561, 1653, ... (23050 of the first 100000 terms);

type 4: products of 3 distinct primes p,q,r such that p*q - 1 = 2*r: 105, 595, 741, ... (22983 of the first 100000 terms);

type 5: products of the cube of a prime p and a distinct prime q such that 2*p^3 + 1 = q: 136, 31375, 3544453, ... (6 of the first 100000 terms);

type 6: products of the cube of a prime p and a distinct prime q such that 2*p^3 - 1 = q: 1431, 1774977571, 12642646591, ... (4 of the first 100000 terms);

type 7: products of the cube of a prime p and a distinct prime q such that p^3 - 1 = 2*q: the only term of this type is 351 = 3^3 * 13.

(No term is a product of the cube of a prime p and a distinct prime q such that p^3 + 1 = 2*q.)

Examples

			Type
(see
cmts)  Initial terms             Notes
-----  ------------------------  -----------------------------
  1    78, 406, 465, ...         p*q*r such that 2*p*q + 1 = r
  2    66, 190, 435, ...         p*q*r such that 2*p*q - 1 = r
  3    231, 561, 1653, ...       p*q*r such that p*q + 1 = 2*r
  4    105, 595, 741, ...        p*q*r such that p*q - 1 = 2*r
  5    136, 31375, 3544453, ...  p^3*q such that 2*p^3 + 1 = q
  6    1431, 1774977571, ...     p^3*q such that 2*p^3 - 1 = q
  7    351 (only)                p^3*q such that p^3 - 1 = 2*q
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A000217 (triangular numbers) and A030626 (8 divisors).

Cf. A063440 (number of divisors of n-th triangular number), A292989 (triangular numbers having exactly 6 divisors).

Programs

Magma

[k:k in [1..16000]| IsSquare(8*k+1) and NumberOfDivisors(k) eq 8]; // Marius A. Burtea, Jan 12 2020

Maple

select(t -> numtheory:-tau(t) = 8, [seq(i*(i+1)/2, i=1..1000)]); # Robert Israel, Jan 13 2020

Mathematica

Select[PolygonalNumber@ Range[180], DivisorSigma[0, #] == 8 &] (* Michael De Vlieger, Jan 11 2020 *)

PARI

isok(k) = ispolygonal(k, 3) && (numdiv(k) == 8); \\ Michel Marcus, Jan 11 2020