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A218443 a(n) = Sum_{k=0..n} floor(n/(3k+2)).

%I A218443 #26 Nov 25 2023 08:03:54
%S A218443 0,0,1,1,2,3,4,4,6,6,8,9,10,10,12,13,15,16,17,17,20,20,22,23,25,26,28,
%T A218443 28,30,31,33,33,36,37,39,41,42,42,44,44,48,49,51,51,54,55,57,58,60,60,
%U A218443 63,64,66,67,68,70,74,74,76,77,80,80,82,82,85,87,89,89,92,93,97,98,100,100,102,103,105,107,109,109
%N A218443 a(n) = Sum_{k=0..n} floor(n/(3k+2)).
%H A218443 Robert Israel, <a href="/A218443/b218443.txt">Table of n, a(n) for n = 0..10000</a>
%H A218443 R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
%F A218443 G.f.: Sum_{k>=0} x^(3*k+2)/((1-x^(3*k+2))*(1-x)). - _Robert Israel_, Feb 28 2017
%F A218443 a(n) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (1 - gamma)/3 = A256843 - (1 - A001620)/3 = -0.0677207... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%p A218443 N:= 100: # to get a(0)..a(N)
%p A218443 A001822:= Vector(N+1):
%p A218443 for m from 2 to N by 3 do
%p A218443   L:= [seq(i,i=m+1..N+1,m)]:
%p A218443   A001822[L]:= map(`+`,A001822[L],1)
%p A218443 od:
%p A218443 ListTools:-PartialSums(convert(A001822,list)); # _Robert Israel_, Feb 28 2017
%t A218443 Table[Sum[Floor[n/(3k+2)],{k,0,n}],{n,0,80}] (* _Harvey P. Dale_, Jun 22 2013 *)
%t A218443 d[n_] := DivisorSum[n, 1 &, Mod[#, 3] == 2 &]; d[0] = 0; Accumulate@Array[d, 100, 0] (* _Amiram Eldar_, Nov 25 2023 *)
%o A218443 (PARI) a(n)=sum(k=0,n\3,(n\(3*k+2)))
%o A218443 (Maxima) A218443[n]:=sum(floor(n/(3*k+2)),k,0,n)$
%o A218443 makelist(A218443[n],n,0,80); /* _Martin Ettl_, Oct 29 2012 */
%Y A218443 Partial sums of A001822.
%Y A218443 Cf. A001620, A218442, A256843.
%K A218443 nonn,easy
%O A218443 0,5
%A A218443 _Benoit Cloitre_, Oct 28 2012