|
%I #10 Jan 14 2020 01:11:25
%S 36,1225,1413721,7885505171090778556470578126753302097454601,
%T 67594562493730400324395236678194231988753004665644702944024074038452164931450549832074201
%N Triangular numbers having exactly 9 divisors.
%C Any number having an odd number of divisors is a square, so each term in this sequence is a term of A001110 (numbers that are both triangular and square). Since A001110(k) = (A000129(k)*A001333(k))^2, A001110(k) will have exactly 9 divisors iff A000129(k) and A001333(k) are both prime (i.e., k is in both A096650 and A099088); the first 5 values of k at which this occurs are 2, 3, 5, 29, and 59.
%C Conjecture: a(5) is the final term of this sequence.
%e Writing the k-th triangular number A000217(k) as T(k):
%e a(1) = T(8) = 8*9/2 = 36 = 2^2 * 3^2;
%e a(2) = T(49) = 49*50/2 = 1225 = 5^2 * 7^2;
%e a(3) = T(1681) = 1681*1682/2 = 1413721 = 29^2 * 41^2.
%e Factorization of larger known terms:
%e a(4) = 44560482149^2 * 63018038201^2;
%e a(5) = 13558774610046711780701^2 * 19175002942688032928599^2.
%Y Intersection of A000217 (triangular numbers) and A030627 (numbers with exactly 9 divisors).
%Y Triangular numbers having exactly k divisors: A068443 (k=4), A292989 (k=6), A330809 (k=8).
%Y Cf. A063440 (number of divisors of n-th triangular number), A242585 (number of divisors of the n-th positive number that is both triangular and square).
%Y Cf. A001110, A000129, A001333, A096650, A099088.
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jan 12 2020
| 