A116607 Sum of the divisors of n which are not divisible by 9.

1, 3, 4, 7, 6, 12, 8, 15, 4, 18, 12, 28, 14, 24, 24, 31, 18, 12, 20, 42, 32, 36, 24, 60, 31, 42, 4, 56, 30, 72, 32, 63, 48, 54, 48, 28, 38, 60, 56, 90, 42, 96, 44, 84, 24, 72, 48, 124, 57, 93, 72, 98, 54, 12, 72, 120, 80, 90, 60, 168, 62, 96, 32, 127, 84, 144, 68, 126, 96

Offset

1,2

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 475 Entry 7(i).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012).

Formula

Expansion of (eta(q^3)^10 / (eta(q) eta(q^9))^3 - 1) / 3 in powers of q.

a(n) is multiplicative with a(3^e) = 4 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.

G.f.: Sum_{k>0} k * x^k /(1 - x^k) - 9*k * x^(9*k) / (1 - x^(9*k)).

L.g.f.: log(Product_{k>=1} (1 - x^(9*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018

Sum_{k=1..n} a(k) ~ (2*Pi^2/27) * n^2. - Amiram Eldar, Oct 04 2022

Example

q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 4*q^9 + ...

Mathematica

With[{c=9Range[20]}, Table[Total[Complement[Divisors[i], c]], {i, 80}]] (* Harvey P. Dale, Dec 19 2010 *)
Drop[CoefficientList[Series[Sum[k * x^k /(1 - x^k) - 9*k * x^(9*k) / (1 - x^(9*k)) , {k, 1, 100}], {x, 0, 100}], x], 1] (* Indranil Ghosh, Mar 25 2017 *)
f[p_, e_] := If[p == 3, 4, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, sigma(n) - if( n%9==0, 9 * sigma(n/9)))}
(PARI) {a(n) = polcoeff( sum( k=1, n, k * (x^k /(1 - x^k) - 9 * x^(9*k) /(1 - x^(9*k))), x * O(x^n)), n)}
(PARI) q='q+O('q^66); Vec( (eta(q^3)^10/(eta(q)*eta(q^9))^3 - 1) /3 ) \\ Joerg Arndt, Mar 25 2017
(Python)
from sympy import divisors
print([sum(i for i in divisors(n) if i%9) for n in range(1, 101)]) # Indranil Ghosh, Mar 25 2017

Crossrefs

A096726(n) = 3*a(n) if n>0.

Cf. A046897, A046913, A113957, A116073, A284326, A284341.

Sequence in context: A366440 A344575 A254981 * A355633 A107749 A353783

Adjacent sequences: A116604 A116605 A116606 * A116608 A116609 A116610

Keyword

nonn,mult

Author

Michael Somos, Feb 19 2006

Status

approved