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

Author

Michael Somos, Feb 12 2011

Keywords

• nonn
• mult

Comments

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

Examples

			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 +...
		

Links

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

Crossrefs

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

Cf. A004016, A005882, A005928.

Programs

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 *)

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 */

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)