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

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, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 97, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 64, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 135, 84, 144, 68, 126, 96

Offset: 1

Author

Jason Earls, May 04 2001

Keywords

• nonn
• easy

Comments

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

Row sums of A161901. - Omar E. Pol, Jan 06 2014

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000203, A037213, A060872, A066839, A070038.

Programs

Maple

A060866 := proc(n)
        numtheory[sigma](n) ;
        if issqr(n) then
                %+sqrt(n) ;
        else
                % ;
        end if;
end proc: # R. J. Mathar, Oct 24 2011

Mathematica

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 *)
Array[If[IntegerQ@ #2, #3 + #2, #3] & @@ {#, Sqrt@ #, DivisorSigma[1, #]} &, 69] (* Michael De Vlieger, Nov 23 2017 *)

PARI

A037213(n) = if(issquare(n,&n),n,0);
A060866(n) = (sigma(n)+A037213(n)); \\ Antti Karttunen, Nov 23 2017, after Jan 25 2003 formula of Vladeta Jovovic

Formula

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

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

Extensions

More terms from Erich Friedman, Jun 03 2001