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%I #17 Mar 20 2022 14:25:45
%S 66,78,105,136,190,231,351,406,435,465,561,595,741,861,903,946,1378,
%T 1431,1653,2211,2278,2485,3081,3655,3741,4371,4465,5151,5253,5995,
%U 6441,7021,7503,8515,8911,9453,9591,10011,10153,10585,11026,12561,13366,14878,15051
%N Triangular numbers having exactly 8 divisors.
%C Terms may be categorized as belonging to the following types:
%C 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);
%C 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);
%C 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);
%C 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);
%C 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);
%C 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);
%C 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.
%C (No term is a product of the cube of a prime p and a distinct prime q such that p^3 + 1 = 2*q.)
%H Robert Israel, <a href="/A330809/b330809.txt">Table of n, a(n) for n = 1..10000</a>
%e Type
%e (see
%e cmts) Initial terms Notes
%e ----- ------------------------ -----------------------------
%e 1 78, 406, 465, ... p*q*r such that 2*p*q + 1 = r
%e 2 66, 190, 435, ... p*q*r such that 2*p*q - 1 = r
%e 3 231, 561, 1653, ... p*q*r such that p*q + 1 = 2*r
%e 4 105, 595, 741, ... p*q*r such that p*q - 1 = 2*r
%e 5 136, 31375, 3544453, ... p^3*q such that 2*p^3 + 1 = q
%e 6 1431, 1774977571, ... p^3*q such that 2*p^3 - 1 = q
%e 7 351 (only) p^3*q such that p^3 - 1 = 2*q
%p select(t -> numtheory:-tau(t) = 8, [seq(i*(i+1)/2, i=1..1000)]); # _Robert Israel_, Jan 13 2020
%t Select[PolygonalNumber@ Range[180], DivisorSigma[0, #] == 8 &] (* _Michael De Vlieger_, Jan 11 2020 *)
%o (PARI) isok(k) = ispolygonal(k, 3) && (numdiv(k) == 8); \\ _Michel Marcus_, Jan 11 2020
%o (Magma) [k:k in [1..16000]| IsSquare(8*k+1) and NumberOfDivisors(k) eq 8]; // _Marius A. Burtea_, Jan 12 2020
%Y Intersection of A000217 (triangular numbers) and A030626 (8 divisors).
%Y Cf. A063440 (number of divisors of n-th triangular number), A292989 (triangular numbers having exactly 6 divisors).
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jan 11 2020
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