A121443 Sum of divisors d of n which are odd and n/d is not divisible by 3.

1, 1, 3, 1, 6, 3, 8, 1, 9, 6, 12, 3, 14, 8, 18, 1, 18, 9, 20, 6, 24, 12, 24, 3, 31, 14, 27, 8, 30, 18, 32, 1, 36, 18, 48, 9, 38, 20, 42, 6, 42, 24, 44, 12, 54, 24, 48, 3, 57, 31, 54, 14, 54, 27, 72, 8, 60, 30, 60, 18, 62, 32, 72, 1, 84, 36, 68, 18, 72, 48, 72, 9, 74, 38, 93, 20, 96, 42

Offset

1,3

Comments

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

References

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 86, Eq. (33.124).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Expansion of c(q) * c(q^2) / 9 where c(q) is a cubic AGM theta function.

Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, -4, ...].

Expansion of (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2)) in powers of q.

Multiplicative with a(2^e) = 1, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 - u*w * (u-2*v) * (v-2*w).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 + 2*u2^3*u3 + 3*u2^2*u3^2 + 6*u1*u2*u3*u6 + 48*u2^2*u6^2 - 3*u1^2*u2*u6 - 3*u1*u2*u3^2 - 24*u2^2*u3*u6 - 30*u1*u2*u6^2. - Michael Somos, Apr 18 2007

G.f.: x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(6*k)))^3 / ((1 - x^k) * (1 - x^(2*k))) = Sum_{k>0} k * x^k * (1 - x^k) / (1 + x^(3*k)).

a(2*n) = a(n), a(2*n + 1) = A185717(n). a(3*n) = 3*a(n). a(6*n + 5) = 6 * A098098(n).

G.f.: Sum_{n = -inf..inf} (-1)^n*x^(3*n+1)/(1 - x^(3*n+1))^2. Cf. A124340. - Peter Bala, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/27 = 0.3655409... (A291050). - Amiram Eldar, Nov 17 2022

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)-3^(-s)+2^(1-s)*3^(-s)). - Amiram Eldar, Jan 03 2023

Example

G.f. = q + q^2 + 3*q^3 + q^4 + 6*q^5 + 3*q^6 + 8*q^7 + q^8 + 9*q^9 + 6*q^10 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ d Mod[ d, 2] Boole[ Mod[ n/d, 3] > 0], {d, Divisors @n}]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]), {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
f[p_, e_] := Which[p == 2, 1, p == 3, p^e, p > 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (d%2) * (n/d%3 > 0) * d))};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)), n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=80) . basis(); A[1] + A[2]; # Michael Somos, Jun 12 2014
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 80); A[2] + A[3]; /* Michael Somos, Jun 12 2014 */

Crossrefs

Cf. A004016, A005882, A005928, A098098, A185717, A291050.

Sequence in context: A158822 A353924 A226132 * A008795 A165188 A294778

Adjacent sequences: A121440 A121441 A121442 * A121444 A121445 A121446

Keyword

nonn,mult

Author

Michael Somos, Jul 30 2006, Apr 18 2007

Status

approved