A279081 Number of divisors of the n-th tetrahedral number.

1, 3, 4, 6, 4, 8, 12, 16, 8, 12, 8, 12, 8, 20, 16, 20, 8, 24, 16, 24, 8, 16, 18, 24, 18, 36, 24, 24, 8, 24, 20, 24, 16, 48, 32, 24, 8, 32, 24, 32, 8, 24, 32, 48, 16, 20, 24, 45, 18, 36, 16, 36, 24, 96, 48, 32, 8, 24, 16, 24, 16, 56, 96, 56, 16, 24, 16, 48, 16

Offset: 1

Author

Jon E. Schoenfield, Jan 06 2017

Keywords

• nonn

Comments

The n-th tetrahedral number is A000292(n) = n*(n+1)*(n+2)/6. The only odd-valued terms are a(1)=1, a(2)=3, and a(48)=45, corresponding to the only nonzero tetrahedral numbers that are also square, i.e., A000292(1)=1, A000292(2)=4, and A000292(48)=19600.

We can write n*(n+1)*(n+2)/6 as the product of three pairwise coprime integers A, B, and C as follows, depending on the value of n mod 12:

.

n mod 12 A B C factor that can be even

======== === ======= ======= =======================

0 n/3 n+1 (n+2)/2 A

1 n (n+1)/2 (n+2)/3 B

2 n/2 (n+1)/3 n+2 C

3 n/3 (n+1)/2 n+2 B

4 n n+1 (n+2)/6 A

5 n (n+1)/6 n+2 B

6 n/6 n+1 n+2 C

7 n (n+1)/2 (n+2)/3 B

8 n (n+1)/3 (n+2)/2 A

9 n/3 (n+1)/2 n+2 B

10 n/2 n+1 (n+2)/3 C

11 n (n+1)/6 n+2 B

.

For all n > 6, A, B, and C are all greater than 1 and share no prime factors, so their product must contain at least three distinct prime factors; consequently, its number of divisors cannot be prime or semiprime. The only semiprimes in this sequence are a(3), a(4), and a(5), and the only prime is a(2).

Examples

			a(48) = tau(48*59*50/6) = tau(19600) = tau(2^4 * 5^2 * 7^2) = (4+1)*(2+1)*(2+1) = 5*3*3 = 45.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000292, A279082, A279083.

Programs

Maple

with(numtheory): seq(tau(n*(n+1)*(n+2)/6), n=1..70) ; # Ridouane Oudra, Jan 25 2024

Mathematica

DivisorSigma[0, Binomial[Range[100]+2, 3]] (* Paolo Xausa, Feb 19 2024 *)

PARI

a(n) = numdiv(n*(n+1)*(n+2)/6); \\ Michel Marcus, Jan 07 2017

Formula

a(n) = A000005(A000292(n)) = A000005(n*(n+1)*(n+2)/6).

From Ridouane Oudra, Jan 25 2024: (Start)

a(6*n) = tau(2*n)*tau(6*n+1)*tau(6*n+2)/2;

a(6*n+1) = tau(6*n+1)*tau(3*n+1)*tau(2*n+1);

a(6*n+2) = tau(6*n+2)*tau(2*n+1)*tau(6*n+4)/2;

a(6*n+3) = tau(2*n+1)*tau(3*n+2)*tau(6*n+5);

a(6*n+4) = tau(6*n+4)*tau(6*n+5)*tau(2*n+2)/2;

a(6*n+5) = tau(6*n+5)*tau(n+1)*tau(6*n+7). (End)