

A331234


Triangular numbers having exactly 9 divisors.


1




OFFSET

1,1


COMMENTS

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.
Conjecture: a(5) is the final term of this sequence.


LINKS

Table of n, a(n) for n=1..5.


EXAMPLE

Writing the kth triangular number A000217(k) as T(k):
a(1) = T(8) = 8*9/2 = 36 = 2^2 * 3^2;
a(2) = T(49) = 49*50/2 = 1225 = 5^2 * 7^2;
a(3) = T(1681) = 1681*1682/2 = 1413721 = 29^2 * 41^2.
Factorization of larger known terms:
a(4) = 44560482149^2 * 63018038201^2;
a(5) = 13558774610046711780701^2 * 19175002942688032928599^2.


CROSSREFS

Intersection of A000217 (triangular numbers) and A030627 (numbers with exactly 9 divisors).
Triangular numbers having exactly k divisors: A068443 (k=4), A292989 (k=6), A330809 (k=8).
Cf. A063440 (number of divisors of nth triangular number), A242585 (number of divisors of the nth positive number that is both triangular and square).
Cf. A001110, A000129, A001333, A096650, A099088.
Sequence in context: A075760 A113938 A001110 * A278806 A064196 A060786
Adjacent sequences: A331231 A331232 A331233 * A331235 A331236 A331237


KEYWORD

nonn


AUTHOR

Jon E. Schoenfield, Jan 12 2020


STATUS

approved



