A116607 - OEIS

%I #46 Oct 04 2022 03:22:28

%S 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,

%T 42,4,56,30,72,32,63,48,54,48,28,38,60,56,90,42,96,44,84,24,72,48,124,

%U 57,93,72,98,54,12,72,120,80,90,60,168,62,96,32,127,84,144,68,126,96

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

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

%H Seiichi Manyama, <a href="/A116607/b116607.txt">Table of n, a(n) for n = 1..10000</a>

%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012).

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

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

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

%F 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

%F Sum_{k=1..n} a(k) ~ (2*Pi^2/27) * n^2. - _Amiram Eldar_, Oct 04 2022

%e 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 + ...

%t With[{c=9Range[20]},Table[Total[Complement[Divisors[i],c]],{i,80}]] (* _Harvey P. Dale_, Dec 19 2010 *)

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, 0, sigma(n) - if( n%9==0, 9 * sigma(n/9)))}

%o (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)}

%o (PARI) q='q+O('q^66); Vec( (eta(q^3)^10/(eta(q)*eta(q^9))^3 - 1) /3 ) \\ _Joerg Arndt_, Mar 25 2017

%o (Python)

%o from sympy import divisors

%o print([sum(i for i in divisors(n) if i%9) for n in range(1, 101)]) # _Indranil Ghosh_, Mar 25 2017

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

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

%K nonn,mult

%O 1,2

%A _Michael Somos_, Feb 19 2006