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A086670
Sum of floor(d/2) where d is a divisor of n.
2
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
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
KEYWORD
nonn
AUTHOR
Jon Perry, Jul 27 2003
STATUS
approved