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A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

%I A078708 #29 Dec 30 2022 03:54:58
%S A078708 1,3,3,7,6,9,8,15,9,18,12,21,14,24,18,31,18,27,20,42,24,36,24,45,31,
%T A078708 42,27,56,30,54,32,63,36,54,48,63,38,60,42,90,42,72,44,84,54,72,48,93,
%U A078708 57,93,54,98,54,81,72,120,60,90,60,126,62,96,72,127,84,108,68,126,72,144
%N A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.
%H A078708 Seiichi Manyama, <a href="/A078708/b078708.txt">Table of n, a(n) for n = 1..10000</a>
%F A078708 G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.
%F A078708 a(n) = (A000203(3*n)-A000203(n))/3. - _Vladeta Jovovic_, Dec 22 2003
%F A078708 G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - _Ilya Gutkovskiy_, Sep 13 2019
%F A078708 From _R. J. Mathar_, May 25 2020: (Start)
%F A078708 a(n) = A326399(n) + A326400(n).
%F A078708 a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)
%F A078708 From _Amiram Eldar_, Oct 30 2022: (Start)
%F A078708 Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.
%F A078708 Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)
%F A078708 Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - _Amiram Eldar_, Dec 30 2022
%t A078708 f[p_, e_] := If[p == 3, 3^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 30 2022 *)
%o A078708 (PARI) for(n=1,70,d=divisors(n); s=0; for(j=1,matsize(d)[2],if((n/d[j])%3>0,s=s+d[j])); print1(s,","))
%o A078708 (PARI) a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)
%Y A078708 Cf. A000203, A035191, A046913, A346933.
%Y A078708 Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).
%Y A078708 Cf. A326399, A326400.
%K A078708 mult,easy,nonn
%O A078708 1,2
%A A078708 _Vladeta Jovovic_, Dec 18 2002
%E A078708 Extended by _Klaus Brockhaus_ and _Benoit Cloitre_, Dec 20 2002