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

%I A279081 #18 Mar 12 2024 15:40:48
%S A279081 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,
%T A279081 24,24,8,24,20,24,16,48,32,24,8,32,24,32,8,24,32,48,16,20,24,45,18,36,
%U A279081 16,36,24,96,48,32,8,24,16,24,16,56,96,56,16,24,16,48,16
%N A279081 Number of divisors of the n-th tetrahedral number.
%C A279081 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.
%C A279081 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:
%C A279081 .
%C A279081 n mod 12   A      B        C     factor that can be even
%C A279081 ========  ===  =======  =======  =======================
%C A279081     0     n/3    n+1    (n+2)/2           A
%C A279081     1      n   (n+1)/2  (n+2)/3             B
%C A279081     2     n/2  (n+1)/3    n+2                 C
%C A279081     3     n/3  (n+1)/2    n+2               B
%C A279081     4      n     n+1    (n+2)/6           A
%C A279081     5      n   (n+1)/6    n+2               B
%C A279081     6     n/6    n+1      n+2                 C
%C A279081     7      n   (n+1)/2  (n+2)/3             B
%C A279081     8      n   (n+1)/3  (n+2)/2           A
%C A279081     9     n/3  (n+1)/2    n+2               B
%C A279081    10     n/2    n+1    (n+2)/3               C
%C A279081    11      n   (n+1)/6    n+2               B
%C A279081 .
%C A279081 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).
%H A279081 Michel Marcus, <a href="/A279081/b279081.txt">Table of n, a(n) for n = 1..10000</a>
%F A279081 a(n) = A000005(A000292(n)) = A000005(n*(n+1)*(n+2)/6).
%F A279081 From _Ridouane Oudra_, Jan 25 2024: (Start)
%F A279081 a(6*n) = tau(2*n)*tau(6*n+1)*tau(6*n+2)/2;
%F A279081 a(6*n+1) = tau(6*n+1)*tau(3*n+1)*tau(2*n+1);
%F A279081 a(6*n+2) = tau(6*n+2)*tau(2*n+1)*tau(6*n+4)/2;
%F A279081 a(6*n+3) = tau(2*n+1)*tau(3*n+2)*tau(6*n+5);
%F A279081 a(6*n+4) = tau(6*n+4)*tau(6*n+5)*tau(2*n+2)/2;
%F A279081 a(6*n+5) = tau(6*n+5)*tau(n+1)*tau(6*n+7). (End)
%e A279081 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.
%p A279081 with(numtheory): seq(tau(n*(n+1)*(n+2)/6), n=1..70) ; # _Ridouane Oudra_, Jan 25 2024
%t A279081 DivisorSigma[0, Binomial[Range[100]+2, 3]] (* _Paolo Xausa_, Feb 19 2024 *)
%o A279081 (PARI) a(n) = numdiv(n*(n+1)*(n+2)/6); \\ _Michel Marcus_, Jan 07 2017
%Y A279081 Cf. A000292, A279082, A279083.
%K A279081 nonn
%O A279081 1,2
%A A279081 _Jon E. Schoenfield_, Jan 06 2017