A006586 a(n) = Sum_{k=1..n} floor((2*n-1)/(2*k+1)). (Formerly M0988)

1, 2, 4, 6, 8, 10, 14, 15, 18, 22, 24, 27, 31, 33, 37, 40, 44, 47, 51, 53, 57, 63, 65, 68, 73, 75, 81, 85, 87, 91, 97, 100, 104, 108, 112, 115, 121, 125, 129, 134, 136, 142, 146, 148, 156, 160, 164, 166, 173, 176, 180, 188, 190, 194, 200, 202, 208, 214, 218, 223, 227

Offset

2,2

Comments

"Nearest integer to" rounds down rather than up when given an exact half-integer.

Can also be described as Sum_{k=1..n-1} { nearest integer to (n-k)/k }. - Jackson Xier, Oct 04 2011

References

Marc LeBrun, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 2..2500

Marc LeBrun, Email to N. J. A. Sloane, Jul 1991

Formula

Clearly a(n) = n log n + O(n); a better estimate is surely possible.

Example

Row sums of the triangle with entries floor((2n-1)/(2k+1)):
   0 : 0;
   1 : 1, 0;
   2 : 1, 1, 0;
   4 : 2, 1, 1, 0;
   6 : 3, 1, 1, 1, 0;
   8 : 3, 2, 1, 1, 1, 0;
  10 : 4, 2, 1, 1, 1, 1, 0;
  14 : 5, 3, 2, 1, 1, 1, 1, 0;
  15 : 5, 3, 2, 1, 1, 1, 1, 1, 0;
  18 : 6, 3, 2, 2, 1, 1, 1, 1, 1, 0;
  22 : 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, 0;
  ...;

Maple

N:=proc(x) local t1; t1:=floor(x); if x-t1 = 1/2 then t1 else floor(x+1/2); fi; end;
f:=n->add(N((n-k)/k), k=1..n-1);
[seq(f(n), n=1..120)]; # N. J. A. Sloane, Oct 02 2011

Mathematica

a[n_]:= Sum[Floor[(2n-1)/(2k+1)], {k, n}]; (* Jackson Xier, Oct 04 2011 *)

Prog

(PARI) a(n)=sum(k=1, n, (2*n-1)\(2*k+1)) \\ Charles R Greathouse IV, Oct 04 2011
(Magma)
A006586:= func< n | (&+[Floor((2*n-1)/(2*k+1)): k in [1..n]]) >;
[A006586(n): n in [2..100]]; // G. C. Greubel, Aug 27 2025
(SageMath)
def A006586(n): return sum((2*n-1)//(2*k+1) for k in range(1, n+1))
print([A006586(n) for n in range(2, 101)]) # G. C. Greubel, Aug 27 2025

Crossrefs

Sequence in context: A253241 A337972 A302979 * A260391 A022292 A225241

Adjacent sequences: A006583 A006584 A006585 * A006587 A006588 A006589

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Gareth McCaughan, Jun 10 2004

Status

approved