OFFSET
1,5
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
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
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Feb 12 2011
STATUS
approved