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
Examples
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 + ...
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
Programs
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Magma
A := Basis( ModularForms( Gamma0(6), 2), 80); A[2] + A[3]; /* Michael Somos, Jun 12 2014 */
-
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 *) -
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
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)).
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
Comments