OFFSET
1,3
COMMENTS
Denominator of the quotient sigma(3*n)/sigma(n). - Labos Elemer, Nov 04 2003
a(n) = b/(3*(c+d)) where b, c, d are the sums of the divisors of 3*n that are congruent respectively to 0, 1 and 2 mod 3. - Michel Lagneau, Nov 05 2012
Sum of powers of 3 dividing n. - Amiram Eldar, Nov 27 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration of A080278 and A080333
FORMULA
G.f.: Sum_{k>=0} 3^k*x^(3^k)/(1-x^(3^k)). - Ralf Stephan, Jun 15 2003
L.g.f.: -log(Product_{k>=0} (1 - x^(3^k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 15 2018
a(n) = sigma(n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = 1 for p != 3.
Dirichlet g.f.: zeta(s) / (1 - 3^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_3(n) + (1/2 + (gamma - 1)/log(3))*n, where gamma is Euler's constant (A001620). (End)
EXAMPLE
a(6) = 4 because the divisors of 3*6 = 18 are {1, 2, 3, 6, 9, 18} => b = 3 + 6 + 9 + 18 = 36, c = 1, d = 2, hence a(6) = b/(3*(c+d)) = 36/(3*(1+2)) = 36/9 = 4. - Michel Lagneau, Nov 05 2012
MATHEMATICA
Table[Denominator[DivisorSigma[1, 3*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := (3^(IntegerExponent[n, 3] + 1) - 1)/2; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)
PROG
(PARI) a(n) = denominator(sigma(3*n)/sigma(n)); \\ Michel Marcus, Dec 15 2019
(PARI) a(n) = (3^(valuation(n, 3) + 1) - 1)/2; \\ Amiram Eldar, Nov 27 2022
CROSSREFS
KEYWORD
nonn,mult,frac
AUTHOR
N. J. A. Sloane, Mar 19 2003
STATUS
approved