A086670 Sum of floor(d/2) where d is a divisor of n.

0, 1, 1, 3, 2, 5, 3, 7, 5, 8, 5, 13, 6, 11, 10, 15, 8, 18, 9, 20, 14, 17, 11, 29, 14, 20, 18, 27, 14, 34, 15, 31, 22, 26, 22, 44, 18, 29, 26, 44, 20, 46, 21, 41, 36, 35, 23, 61, 27, 45, 34, 48, 26, 58, 34, 59, 38, 44, 29, 82, 30, 47, 49, 63, 40, 70, 33, 62, 46, 70, 35, 96, 36, 56

Offset: 1

Author

Jon Perry, Jul 27 2003

Keywords

• nonn

Comments

Inverse Mobius transform of A004526. - R. J. Mathar, Jan 19 2009

Examples

			10 has divisors 1,2,5,10. floor(d/2) gives 0,1,2,5, therefore a(10)=8.
		

Crossrefs

Cf. A000203, A001227, A004526, A079247.

Cf. A135539.

Programs

Mathematica

Table[Total[Floor[Divisors[n]/2]],{n,80}] (* Harvey P. Dale, Feb 13 2023 *)

PARI

for (n=1,100,s=0; fordiv(i=n,i,s+=floor(i/2)); print1(","s))

PARI

a(n) = my(f = factor(n)); (sigma(f) - (numdiv(f)/(valuation(n, 2)+1)))>>1 \\ David A. Corneth, Apr 15 2022 using Franklin T. Adams-Watters's formula

Formula

G.f.: Sum_{n>=1} floor(n/2)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011

a(n) = (A000203(n) - A001227(n)) / 2. - Franklin T. Adams-Watters, Jan 05 2012

G.f.: Sum_{k>=1} x^(2*k) / ((1 + x^k) * (1 - x^k)^2). - Ilya Gutkovskiy, Aug 02 2021

a(n) = Sum_{i=1..floor(n/2)} A135539(n,2*i). - Ridouane Oudra, Apr 15 2022