A323747 Smallest triangular number whose number of divisors is the n-th triangular number, or 0 if no such number exists.

1, 0, 28, 496, 1631432881, 0, 8256, 2016, 41616, 0, 169878528, 2717872128, 0

Offset: 1

Author

Jon E. Schoenfield, May 25 2019

Keywords

• nonn
• more

Comments

Additional known terms include a(15)=270480, a(16)=77309214720, a(19)=117261433825538425475625, a(20)=7874496, a(22)=0, a(23)=316659361382400, a(24)=100472400, a(25)=0, a(27)=18951806016, a(28)=35184372088827805696000000, a(31)=20752587086144471040, a(32)=3877678080.

It is known (see the comments and links at A081978) that a(n)=0 for every n such that n*(n+1)/2 is an odd composite not divisible by 3; this includes n = 10, 13, 22, 25, ..., i.e., all n such that n mod 12 is 1 or 10.

Conjectures:

1. a(n) > 0 for every n such that n*(n+1)/2 is even.

2. a(n) = 0 for every n such that n*(n+1)/2 is odd except n = 1, 5, and 9 (whose corresponding values of n*(n+1)/2 are 1, 15, and 45, respectively). Can this be proved for any of the values of n in {14, 17, 18, 21, 26, 29, 30}?

Examples

			a(1) = 1 because 1 is the only triangular number having A000217(1)=1 divisors.
a(2) = 0 because no triangular number has A000217(2)=3 divisors. (Each number with 3 divisors is the square of a prime, and no such number can be of the form k*(k+1)/2.)
a(3) = 28 because 28 = 7*(7+1)/2 = 2^2 * 7 is the smallest triangular number with A000217(3)=6 divisors.
a(5) = 1631432881 = 13^4 * 239^2 is the only triangular with A000217(5)=15 divisors.
		

Crossrefs

Cf. A000217, A081978, A116541, A242585, A292989.

Extensions

a(6)-a(13) and updated comments from Jon E. Schoenfield, Jan 29 2021