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A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.

%I A060866 #27 Dec 15 2017 17:35:00
%S A060866 2,3,4,9,6,12,8,15,16,18,12,28,14,24,24,35,18,39,20,42,32,36,24,60,36,
%T A060866 42,40,56,30,72,32,63,48,54,48,97,38,60,56,90,42,96,44,84,78,72,48,
%U A060866 124,64,93,72,98,54,120,72,120,80,90,60,168,62,96,104,135,84,144,68,126,96
%N A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.
%C A060866 Paraphrasing the Jovovic formula: if n is not a square then a(n) = sigma(n), the sum of divisors of n, otherwise a(n) = sigma(n) + sqrt(n). - _Omar E. Pol_, Jun 23 2009
%C A060866 Row sums of A161901. - _Omar E. Pol_, Jan 06 2014
%H A060866 Antti Karttunen, <a href="/A060866/b060866.txt">Table of n, a(n) for n = 1..16384</a>
%F A060866 a(n) = A066839(n)+A070038(n) = A000203(n)+A037213(n). G.f.: Sum_{n>0} n*x^n*(x^(n*(n-1))-x^(n^2)+1)/(1-x^n). - _Vladeta Jovovic_, Jan 25 2003
%F A060866 a(n) = sum_{i=1..floor(sqrt(n))} (n+i^2)*(1-ceiling(n/i)+floor(n/i))/i. - _Wesley Ivan Hurt_, Jul 14 2014
%e A060866 a(4)=9 because pairs of factors are 1*4 and 2*2 and 1+4+2+2=9. a(6)=12 because pairs of factors are 1*6 and 2*3 and 1+6+2+3=12.
%p A060866 A060866 := proc(n)
%p A060866         numtheory[sigma](n) ;
%p A060866         if issqr(n) then
%p A060866                 %+sqrt(n) ;
%p A060866         else
%p A060866                 % ;
%p A060866         end if;
%p A060866 end proc: # _R. J. Mathar_, Oct 24 2011
%t A060866 Table[Sum[(i^2 + n) (1 - Ceiling[n/i] + Floor[n/i])/i, {i, Floor[Sqrt[n]]}], {n, 100}] (* _Wesley Ivan Hurt_, Jul 14 2014 *)
%t A060866 Array[If[IntegerQ@ #2, #3 + #2, #3] & @@ {#, Sqrt@ #, DivisorSigma[1, #]} &, 69] (* _Michael De Vlieger_, Nov 23 2017 *)
%o A060866 (PARI)
%o A060866 A037213(n) = if(issquare(n,&n),n,0);
%o A060866 A060866(n) = (sigma(n)+A037213(n)); \\ _Antti Karttunen_, Nov 23 2017, after Jan 25 2003 formula of _Vladeta Jovovic_
%Y A060866 Cf. A000203, A037213, A060872, A066839, A070038.
%K A060866 nonn,easy
%O A060866 1,1
%A A060866 _Jason Earls_, May 04 2001
%E A060866 More terms from _Erich Friedman_, Jun 03 2001