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

1,4

Comments

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

Links

Table of n, a(n) for n=1..74.

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000203, A001227, A004526, A079247.

Cf. A135539.

Sequence in context: A083242 A111618 A107128 * A075888 A075889 A181771

Adjacent sequences: A086667 A086668 A086669 * A086671 A086672 A086673

Keyword

nonn

Author

Jon Perry, Jul 27 2003

Status

approved