This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330809 #17 Mar 20 2022 14:25:45 %S A330809 66,78,105,136,190,231,351,406,435,465,561,595,741,861,903,946,1378, %T A330809 1431,1653,2211,2278,2485,3081,3655,3741,4371,4465,5151,5253,5995, %U A330809 6441,7021,7503,8515,8911,9453,9591,10011,10153,10585,11026,12561,13366,14878,15051 %N A330809 Triangular numbers having exactly 8 divisors. %C A330809 Terms may be categorized as belonging to the following types: %C A330809 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 A330809 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 A330809 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 A330809 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 A330809 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 A330809 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 A330809 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 A330809 (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 A330809 Robert Israel, <a href="/A330809/b330809.txt">Table of n, a(n) for n = 1..10000</a> %e A330809 Type %e A330809 (see %e A330809 cmts) Initial terms Notes %e A330809 ----- ------------------------ ----------------------------- %e A330809 1 78, 406, 465, ... p*q*r such that 2*p*q + 1 = r %e A330809 2 66, 190, 435, ... p*q*r such that 2*p*q - 1 = r %e A330809 3 231, 561, 1653, ... p*q*r such that p*q + 1 = 2*r %e A330809 4 105, 595, 741, ... p*q*r such that p*q - 1 = 2*r %e A330809 5 136, 31375, 3544453, ... p^3*q such that 2*p^3 + 1 = q %e A330809 6 1431, 1774977571, ... p^3*q such that 2*p^3 - 1 = q %e A330809 7 351 (only) p^3*q such that p^3 - 1 = 2*q %p A330809 select(t -> numtheory:-tau(t) = 8, [seq(i*(i+1)/2, i=1..1000)]); # _Robert Israel_, Jan 13 2020 %t A330809 Select[PolygonalNumber@ Range[180], DivisorSigma[0, #] == 8 &] (* _Michael De Vlieger_, Jan 11 2020 *) %o A330809 (PARI) isok(k) = ispolygonal(k, 3) && (numdiv(k) == 8); \\ _Michel Marcus_, Jan 11 2020 %o A330809 (Magma) [k:k in [1..16000]| IsSquare(8*k+1) and NumberOfDivisors(k) eq 8]; // _Marius A. Burtea_, Jan 12 2020 %Y A330809 Intersection of A000217 (triangular numbers) and A030626 (8 divisors). %Y A330809 Cf. A063440 (number of divisors of n-th triangular number), A292989 (triangular numbers having exactly 6 divisors). %K A330809 nonn %O A330809 1,1 %A A330809 _Jon E. Schoenfield_, Jan 11 2020