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A279081
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Number of divisors of the n-th tetrahedral number.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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:
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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
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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).
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LINKS
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FORMULA
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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)
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EXAMPLE
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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.
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MAPLE
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with(numtheory): seq(tau(n*(n+1)*(n+2)/6), n=1..70) ; # Ridouane Oudra, Jan 25 2024
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MATHEMATICA
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DivisorSigma[0, Binomial[Range[100]+2, 3]] (* Paolo Xausa, Feb 19 2024 *)
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PROG
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(PARI) a(n) = numdiv(n*(n+1)*(n+2)/6); \\ Michel Marcus, Jan 07 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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