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Sum of divisors d of n such that n/d is not congruent to 0 mod 3.
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%I #29 Dec 30 2022 03:54:58

%S 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 42,27,56,30,54,32,63,36,54,48,63,38,60,42,90,42,72,44,84,54,72,48,93,

%U 57,93,54,98,54,81,72,120,60,90,60,126,62,96,72,127,84,108,68,126,72,144

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

%H Seiichi Manyama, <a href="/A078708/b078708.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.

%F a(n) = (A000203(3*n)-A000203(n))/3. - _Vladeta Jovovic_, Dec 22 2003

%F G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - _Ilya Gutkovskiy_, Sep 13 2019

%F From _R. J. Mathar_, May 25 2020: (Start)

%F a(n) = A326399(n) + A326400(n).

%F a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)

%F From _Amiram Eldar_, Oct 30 2022: (Start)

%F Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)

%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - _Amiram Eldar_, Dec 30 2022

%t 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 (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 (PARI) a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)

%Y Cf. A000203, A035191, A046913, A346933.

%Y Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).

%Y Cf. A326399, A326400.

%K mult,easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Dec 18 2002

%E Extended by _Klaus Brockhaus_ and _Benoit Cloitre_, Dec 20 2002