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%I #41 Dec 30 2022 03:56:10
%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
%F 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
%F From _Amiram Eldar_, Dec 30 2022: (Start)
%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2/2^s)*(1-3/3^s).
%F Sum_{k=1..n} a(k) ~ c*n^2, where c = Pi^2/36 = 0.274155... (A353908). (End)
%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}] (* _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 /* _Paul D. Hanna_, Feb 17 2013 */
%Y Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104, A353908.
%Y Cf. A004016, A005882, A005928.
%K nonn,mult
%O 1,5
%A _Michael Somos_, Feb 12 2011
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