A080278 a(n) = (3^(v_3(n) + 1) - 1)/2, where v_3(n) = highest power of 3 dividing n = A007949(n).

1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 40, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1, 121, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 1, 4, 1, 1, 4, 1, 1

Offset: 1

Author

N. J. A. Sloane, Mar 19 2003

Keywords

• nonn
• mult
• frac

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

Examples

			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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Klaus Brockhaus, Illustration of A080278 and A080333

Crossrefs

Cf. A000203, A001620, A007949, A080333, A088838 (numerator of sigma(3*n)/sigma(n)).

Programs

Maple

A080278 := n->(3^(A007949(n)+1)-1)/2;

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 *)

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

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)