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A116607 Sum of the divisors of n which are not divisible by 9. 10
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 (list; graph; refs; listen; history; text; internal format)
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

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}]] [From 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 *)

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>0]) for n in xrange(1, 101)] # Indranil Ghosh, Mar 25 2017

CROSSREFS

A097626(n) = 3*a(n) if n>0.

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

Sequence in context: A049418 A051378 A254981 * A107749 A093811 A088000

Adjacent sequences:  A116604 A116605 A116606 * A116608 A116609 A116610

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Feb 19 2006

STATUS

approved

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Last modified January 20 04:14 EST 2019. Contains 319323 sequences. (Running on oeis4.)