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A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

%I A184389 #52 May 30 2025 10:16:25
%S A184389 1,3,3,6,3,10,3,10,6,10,3,21,3,10,10,15,3,21,3,21,10,10,3,36,6,10,10,
%T A184389 21,3,36,3,21,10,10,10,45,3,10,10,36,3,36,3,21,21,10,3,55,6,21,10,21,
%U A184389 3,36,10,36,10,10,3,78,3,10,21,28,10,36,3,21,10,36,3,78
%N A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).
%C A184389 Length of row n in triangle A187207. - _Omar E. Pol_, Aug 07 2011
%C A184389 Number of pairs of even divisors of 2n, (d1,d2), such that d1<=d2. - _Wesley Ivan Hurt_, Aug 24 2020
%H A184389 Enrique Pérez Herrero, <a href="/A184389/b184389.txt">Table of n, a(n) for n = 1..5000</a>
%F A184389 a(n) = A000217(A000005(n)) = (1/2)*A000005(n)*(A000005(n)+1).
%F A184389 a(n) = A066446(n) + A000005(n) = A035116(n) - A066446(n). - _Reinhard Zumkeller_, Sep 08 2015
%F A184389 Dirichlet g.f.: zeta(s)^2*(zeta(s)^2 + zeta(2*s))/(2*zeta(2*s)). - _Ilya Gutkovskiy_, Jun 25 2016
%F A184389 a(n) = Sum_{d1|(2*n), d2|(2*n), d1 and d2 even, d1<=d2} 1. - _Wesley Ivan Hurt_, Aug 24 2020
%F A184389 a(n) = Sum_{d|n} A018892(d). - _Daniel Suteu_, Jan 08 2021
%F A184389 a(n) = Sum_{d|n} A135539(n,d). - _Ridouane Oudra_, May 29 2025
%F A184389 a(n) = A337362(n) + A129308(n). - _Ridouane Oudra_, May 30 2025
%e A184389 For n = 4; tau(4) = 3; a(4) = 1+2+3 = 6.
%p A184389 A184389 := proc(n)
%p A184389     A000217(numtheory[tau](n)) ;
%p A184389 end proc: # _R. J. Mathar_, Oct 04 2014
%t A184389 ((#+1)#)/2&/@DivisorSigma[0,Range[80]] (* _Harvey P. Dale_, Feb 27 2013 *)
%o A184389 (Haskell)
%o A184389 a184389 = a000217 . a000005'  -- _Reinhard Zumkeller_, Sep 08 2015
%o A184389 (PARI) a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ _Michel Marcus_, Jun 25 2016
%Y A184389 Cf. A000005 (tau), A000217 (triangular numbers).
%Y A184389 Cf. A184387, A184388, A184390, A184391, A130674.
%Y A184389 Cf. A035116, A066446.
%Y A184389 Cf. A187207, A135539, A337362, A129308.
%K A184389 nonn
%O A184389 1,2
%A A184389 _Jaroslav Krizek_, Jan 12 2011