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A078181 a(n) = Sum_{d|n, d == 1 (mod 3)} d.

%I A078181 #37 Nov 26 2023 03:05:34
%S A078181 1,1,1,5,1,1,8,5,1,11,1,5,14,8,1,21,1,1,20,15,8,23,1,5,26,14,1,40,1,
%T A078181 11,32,21,1,35,8,5,38,20,14,55,1,8,44,27,1,47,1,21,57,36,1,70,1,1,56,
%U A078181 40,20,59,1,15,62,32,8,85,14,23,68,39,1,88,1,5,74,38,26,100,8,14,80,71,1
%N A078181 a(n) = Sum_{d|n, d == 1 (mod 3)} d.
%H A078181 Seiichi Manyama, <a href="/A078181/b078181.txt">Table of n, a(n) for n = 1..10000</a>
%F A078181 G.f.: Sum_{n>=0} (3*n+1)*x^(3*n+1)/(1-x^(3*n+1)).
%F A078181 G.f.: -q*P'/P where P = Product_{n>=0} (1 - q^(3*n+1)). - _Joerg Arndt_, Aug 03 2011
%F A078181 Conjecture. If a(n)=n+1 then n==1 (mod 3). (Is this easy to settle? It has been verified for n=1,2,3,...,2000.) - _John W. Layman_, Apr 03 2006
%F A078181 The conjecture is false. The first and only counterexample below 10^8 is a(6800) = 6801 and 6800 == 2 (mod 3). - Lambert Herrgesell (zero815(AT)googlemail.com), May 06 2008
%F A078181 Equals A051731 * [1, 0, 0, 4, 0, 0, 7, 0, 0, 10, ...]. - _Gary W. Adamson_, Nov 06 2007
%F A078181 A272027(n/3) + a(n) + A078182(n) = A000203(n). - _R. J. Mathar_, May 25 2020
%F A078181 G.f.: Sum_{n >= 1} x^n*(1 + 2*x^(3*n))/(1 - x^(3*n))^2. - _Peter Bala_, Dec 19 2021
%F A078181 Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 = 0.274155... (A353908). - _Amiram Eldar_, Nov 26 2023
%p A078181 A078181 := proc(n)
%p A078181     a := 0 ;
%p A078181     for d in numtheory[divisors](n) do
%p A078181         if modp(d,3) =1 then
%p A078181             a :=a+d ;
%p A078181         end if;
%p A078181     end do:
%p A078181     a;
%p A078181 end proc: # _R. J. Mathar_, May 11 2016
%t A078181 a[n_] := Plus @@ Select[Divisors[n], Mod[#, 3] == 1 &]; Array[a, 100] (* _Giovanni Resta_, May 11 2016 *)
%Y A078181 Cf. A001817, A001822, A035382, A051731, A272716, A353908.
%Y A078181 Cf. Sum_{d|n, d==1 mod k} d: A000593 (k=2), this sequence (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).
%K A078181 easy,nonn
%O A078181 1,4
%A A078181 _Vladeta Jovovic_, Nov 21 2002