login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A186099 Sum of divisors of n congruent to 1 or 5 mod 6. 2

%I

%S 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,

%T 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,

%U 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

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

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

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

%F 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

%F 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

%e 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 +...

%e 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 +...

%e 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 +...

%t Table[Total[Select[Divisors[n],MemberQ[{1,5},Mod[#,6]]&]],{n,0,100}] (* From _Harvey P. Dale_, Feb 24 2011 *)

%o (PARI) {a(n) = sumdiv( n, d, d * (1 == gcd( d, 6) ) )}

%o (PARI) {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]}

%o (PARI) a(n)=sigma(n/2^valuation(n,2)/3^valuation(n,3)) \\ _Charles R Greathouse IV_, Dec 07 2011

%o (PARI)

%o {S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}

%o {a(n)=n*polcoeff(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n),n)}

%o for(n=1,80,print1(a(n),", "))

%o /* From _Paul D. Hanna_, Feb 17 2013 */

%Y Cf. A000593, A046913, A113957, A116073, A003105.

%K nonn,mult

%O 1,5

%A _Michael Somos_, Feb 12 2011

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 22 16:00 EST 2014. Contains 252364 sequences.