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 A186099 Sum of divisors of n congruent to 1 or 5 mod 6. 8

%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.

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

%H Seiichi Manyama, <a href="/A186099/b186099.txt">Table of n, a(n) for n = 1..10000</a>

%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

%F a(n) = A284098(n) + A284104(n). - _Seiichi Manyama_, Mar 24 2017

%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 *)

%t a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* _Michael Somos_, Jun 27 2017 *)

%t a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* _Michael Somos_, Jun 27 2017 *)

%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, A284098, A284104.

%K nonn,mult

%O 1,5

%A _Michael Somos_, Feb 12 2011

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