A096726 -id:A096726 - cp's OEIS frontend

A033686 One-ninth of theta series of A2[hole]^2.

1, 2, 5, 4, 8, 6, 14, 8, 14, 10, 21, 16, 20, 14, 28, 16, 31, 18, 40, 20, 32, 28, 42, 24, 38, 32, 62, 28, 44, 30, 56, 40, 57, 34, 70, 36, 72, 38, 70, 48, 62, 52, 85, 44, 68, 46, 112, 56, 74, 50, 100, 64, 80, 64, 98, 56, 108, 58, 124, 60, 112, 76, 112, 64, 98, 66, 155, 80, 104

Offset: 0

N. J. A. Sloane

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Number of partition pairs of n where each partition is 3-core (see Theorem 2.1 of Wang link). - Michel Marcus, Jul 14 2015

			G.f. = 1 + 2*x + 5*x^2 + 4*x^3 + 8*x^4 + 6*x^5 + 14*x^6 + 8*x^7 + 14*x^8 + ...
G.f. = q^2 + 2*q^5 + 5*q^8 + 4*q^11 + 8*q^14 + 6*q^17 + 14*q^20 + 8*q^23 + ...
Theta series of A2[hole]^2 = c(q)^2 = 9*q^(2/3) + 18*q^(5/3) + 45*q^(8/3) + 36*q^(11/3) + 72*q^(14/3) + 54*q^(17/3) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111, Eq (63)^2.

Muniru A Asiru, Table of n, a(n) for n = 0..20000
Kevin B. Ford, Some infinite series identities, Proc. Amer. Math. Soc., Volume 119, Number 3 (November 1993), 1019-1020.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A033687, A096726, A097723, A134079, A144615, A242874, A281722.

Cf. A000203 (sigma), A016789 (3n+2).

sequence := List([1..100010],n->Sigma(3*n-1)/3); # Muniru A Asiru, Dec 29 2017

Basis( ModularForms( Gamma0(9), 2), 195)[3]; /* Michael Somos, Jul 14 2015 */

[SumOfDivisors(3*n+2)/3: n in [0..70]]; // Vincenzo Librandi, Jan 13 2018

with(numtheory): seq(sigma(3*n-1)/3, n=1..2000); # Muniru A Asiru, Jan 18 2018

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 26 2014 *)
Array[ DivisorSigma[1, 3 # - 1]/3 &, 69] (* Robert G. Wilson v, Jan 12 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Oct 17 2006 */

a(n)=sigma(3*n+2)/3; \\ Michel Marcus, Jul 14 2015

ModularForms( Gamma0(9), 2, prec=195).2 # Michael Somos, May 26 2014

a(n) = sigma(3*n+2)/3. Euler transform of period 3 sequence [2, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of q^(-2/3) * c(q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function. - Michael Somos, Oct 17 2006
Expansion of q^(-2/3) * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Mar 16 2012
Convolution square of A033687. - Michael Somos, Oct 17 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A242874.
27 * a(n) = A096726(3*n + 2) - A281722(3*n + 2). - Michael Somos, Sep 04 2017
a(n) = A144615(n)/3. - Robert G. Wilson v, Jan 12 2018
From Peter Bala, Jan 07 2021: (Start)
a(n) = (-1)^n*A134079(n).
A(x) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(3*n+1))^2 = Sum_{n = -oo..oo} x^(4*n+2)/(1 - x^(3*n+2))^2 (apply Ford, equation 1, with c = x^(3/2), d = x^(1/2), |x| < 1 to the g.f. Sum_{n = -oo..oo} x^n /(1 - x^(3*n + 1)) of A033687).
Conjectural g.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(3*n+2))^2 = Sum_{n = -oo..oo} x^(5*n+1)/(1 - x^(3*n+1))^2. (End)
Sum_{k=1..n} a(k) = (2*Pi^2/27) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

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

Original entry on oeis.org

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

Michael Somos, Feb 19 2006

			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 + ...

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

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).

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

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

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 *)

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

{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)}

q='q+O('q^66); Vec( (eta(q^3)^10/(eta(q)*eta(q^9))^3 - 1) /3 ) \\ Joerg Arndt, Mar 25 2017

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

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

A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140

Offset: 0

Michael Somos, Jul 02 2013

			G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - _Antti Karttunen_, Nov 23 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to sums of divisors.

Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.

A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};

{a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};

A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];

a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022

More terms from Antti Karttunen, Nov 23 2017

A281722 Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

Original entry on oeis.org

1, 3, -18, 12, 21, -36, 36, 24, -90, 12, 54, -72, 84, 42, -144, 72, 93, -108, 36, 60, -252, 96, 108, -144, 180, 93, -252, 12, 168, -180, 216, 96, -378, 144, 162, -288, 84, 114, -360, 168, 270, -252, 288, 132, -504, 72, 216, -288, 372, 171, -558, 216, 294, -324

Offset: 0

Michael Somos, Jan 28 2017

sign

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

			G.f. = 1 + 3*q - 18*q^2 + 12*q^3 + 21*q^4 - 36*q^5 + 36*q^6 + 24*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from G. C. Greubel)

Cf. A004016, A005928, A033686, A096726, A144614.

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 (QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3]^2, {q, 0, n}];

{a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A)^3 * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A)^2, n))};

Convolution of the sequences A004016 and A005928.
The g.f. is the product of the g.f.'s for A004016 and A005928. - N. J. A. Sloane, Jan 30 2017
Expansion of eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144614.
a(3*n + 2) = A096726(3*n + 2) - 27 * A033686(n). a(n) == A096726(n) (mod 27). - Michael Somos, Sep 04 2017

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A033686 One-ninth of theta series of A2[hole]^2.

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

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A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

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A281722 Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

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