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Expansion of Sum_{k>=1} k * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)).
1

%I #33 Nov 06 2022 03:16:49

%S 1,2,3,4,4,6,8,8,9,8,10,12,14,16,12,16,16,18,20,16,24,20,22,24,21,28,

%T 27,32,28,24,32,32,30,32,32,36,38,40,42,32,40,48,44,40,36,44,46,48,57,

%U 42,48,56,52,54,40,64,60,56,58,48,62,64,72,64,56,60

%N Expansion of Sum_{k>=1} k * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)).

%H Amiram Eldar, <a href="/A326575/b326575.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, n/d==1 (mod 6)} d - Sum_{d|n, n/d==5 (mod 6)} d.

%F G.f.: Sum_{k>=0} x^(6*k+1) / (1 - x^(6*k+1))^2 - x^(6*k+5) / (1 - x^(6*k+5))^2. - _Michael Somos_, Oct 23 2019

%F Multiplicative with a(p^e) = p^e if p < 5, (p^(e+1)-(-1)^(e+1))/(p+1) if p == 5 (mod 6), and (p^(e+1)-1)/(p-1) if p == 1 (mod 6). - _Amiram Eldar_, Dec 02 2020

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 5 (mod 6)} 1/(1+1/p^2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) = A340578 * A175646 / 2 = 0.48831400806... . - _Amiram Eldar_, Nov 06 2022

%e G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ... - _Michael Somos_, Oct 23 2019

%t nmax = 66; CoefficientList[Series[Sum[k x^k (1 + x^(2 k))/(1 + x^(2 k) + x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 6]] &] - DivisorSum[n, # &, MemberQ[{5}, Mod[n/#, 6]] &], {n, 1, 66}]

%t f[p_, e_] := Which[p < 5, p^e, Mod[p, 6] == 5, (p^(e + 1) - (-1)^(e + 1))/(p + 1), Mod[p, 6] == 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 02 2020 *)

%o (PARI) a(n) = { sumdiv(n, d, d*((n/d%6==1)-(n/d%6==5))) } \\ _Andrew Howroyd_, Sep 12 2019

%o (PARI) {a(n) = if( n<0, 0, sumdiv( n, d, n/d * kronecker( -12, d)))}; /* _Michael Somos_, Oct 23 2019 */

%Y Cf. A003586 (fixed points), A035178, A050469, A122373, A326401.

%Y Cf. A175646, A340578.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Sep 12 2019