A186099 Sum of divisors of n congruent to 1 or 5 mod 6.

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 31, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 57, 31, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38, 31, 20, 96, 14, 80, 6

Offset

1,5

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

Expansion of (1 + a(x)^2 - 2*a(x^2)^2) / 12 in powers of x where a() is a cubic AGM function.

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

Equals the logarithmic derivative of A003105, where A003105(n) = number of partitions of n into parts 6*n+1 or 6*n-1. - Paul D. Hanna, Feb 17 2013

L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} S(n,x)*x^n/n where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013

a(n) = A284098(n) + A284104(n). - Seiichi Manyama, Mar 24 2017

G.f.: Sum_{n >= 1} x^n*(x^(10*n) + 5*x^(6*n) + 5*x^(4*n) + 1)/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021

From Amiram Eldar, Dec 30 2022: (Start)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2/2^s)*(1-3/3^s).

Sum_{k=1..n} a(k) ~ c*n^2, where c = Pi^2/36 = 0.274155... (A353908). (End)

Example

G.f.: x + x^2 + x^3 + x^4 + 6*x^5 + x^6 + 8*x^7 + x^8 + x^9 + 6*x^10 + 12*x^11 +...
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 +...
where exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 +...+ A003105(n)*x^n +...

Mathematica

Table[Total[Select[Divisors[n], MemberQ[{1, 5}, Mod[#, 6]]&]], {n, 0, 100}]  (* Harvey P. Dale, Feb 24 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* Michael Somos, Jun 27 2017 *)
a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)

Prog

(PARI) {a(n) = sumdiv( n, d, d * (1 == gcd(d, 6) ))};
(PARI) {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]};
(PARI) a(n)=sigma(n/2^valuation(n, 2)/3^valuation(n, 3)) \\ Charles R Greathouse IV, Dec 07 2011
(PARI)
{S(n, x)=sumdiv(n, d, d*(1-x^d)^(n/d))}
{a(n)=n*polcoeff(sum(k=1, n, S(k, x)*x^k/k)+x*O(x^n), n)}
for(n=1, 80, print1(a(n), ", "))
/* Paul D. Hanna, Feb 17 2013 */

Crossrefs

Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104, A353908.

Cf. A004016, A005882, A005928.

Sequence in context: A156921 A094214 A001622 * A021622 A073228 A256853

Adjacent sequences: A186096 A186097 A186098 * A186100 A186101 A186102

Keyword

nonn,mult

Author

Michael Somos, Feb 12 2011

Status

approved