A060831 a(n) = Sum_{k=1..n} (number of odd divisors of k) (cf. A001227).

0, 1, 2, 4, 5, 7, 9, 11, 12, 15, 17, 19, 21, 23, 25, 29, 30, 32, 35, 37, 39, 43, 45, 47, 49, 52, 54, 58, 60, 62, 66, 68, 69, 73, 75, 79, 82, 84, 86, 90, 92, 94, 98, 100, 102, 108, 110, 112, 114, 117, 120, 124, 126, 128, 132, 136, 138, 142, 144, 146, 150, 152, 154, 160

Offset: 0

Author

Henry Bottomley, May 01 2001

Keywords

• nonn

Comments

The old definition was "Number of sums less than or equal to n of sequences of consecutive positive integers (including sequences of length 1)."

In other words, a(n) is also the total number of partitions of all positive integers <= n into consecutive parts, n >= 1. - Omar E. Pol, Dec 03 2020

Starting with 1 = row sums of triangle A168508. - Gary W. Adamson, Nov 27 2009

The subsequence of primes in this sequence begins, through a(100): 2, 5, 7, 11, 17, 19, 23, 29, 37, 43, 47, 73, 79, 173, 181, 223, 227, 229, 233, 263. - Jonathan Vos Post, Feb 13 2010

Apart from the initial zero, a(n) is also the total number of subparts of the symmetric representations of sigma of all positive integers <= n. Hence a(n) is also the total number of subparts in the terraces of the stepped pyramid with n levels described in A245092. For more information see A279387 and A237593. - Omar E. Pol, Dec 17 2016

a(n) is also the total number of partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, May 14 2017

Zero together with the row sums of A235791. - Omar E. Pol, Dec 18 2020

Examples

			E.g., for a(7), we consider the odd divisors of 1,2,3,4,5,6,7, which gives 1,1,2,1,2,2,2 = 11. - _Jon Perry_, Mar 22 2004
Example illustrating the old definition: a(7) = 11 since 1, 2, 3, 4, 5, 6, 7, 1+2, 2+3, 3+4, 1+2+3 are all 7 or less.
From _Omar E. Pol_, Dec 02 2020: (Start)
Illustration of initial terms:
                              Diagram
   n   a(n)
   0     0                          _|
   1     1                        _|1|
   2     2                      _|1 _|
   3     4                    _|1  |1|
   4     5                  _|1   _| |
   5     7                _|1    |1 _|
   6     9              _|1     _| |1|
   7    11            _|1      |1  | |
   8    12          _|1       _|  _| |
   9    15        _|1        |1  |1 _|
  10    17      _|1         _|   | |1|
  11    19    _|1          |1   _| | |
  12    21   |1            |   |1  | |
...
a(n) is also the total number of horizontal line segments in the first n levels of the diagram. For n = 5 there are seven horizontal line segments, so a(5) = 7. Cf. A237048, A286001. (End)
From _Omar E. Pol_, Dec 19 2020: (Start)
a(n) is also the number of regions in the diagram of the symmetries of sigma after n stages, including the subparts, as shown below (Cf. A279387):
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _| |_|_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.  0    1      2        4          5            7              9
(End)
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Zero together with the partial sums of A001227.

Cf. A000005, A001620, A006218, A168508, A235791, A236104, A237048, A237590, A237593, A245092, A279387, A286001.

Programs

Maple

A060831 := proc(n)
    add(numtheory[tau](n-i+1),i=1..ceil(n/2)) ;
end proc:
seq(A060831(n),n=0..100) ; # Wesley Ivan Hurt, Oct 02 2013

Mathematica

f[n_] := Sum[ -(-1^k)Floor[n/(2k - 1)], {k, n}]; Table[ f[n], {n, 0, 65}] (* Robert G. Wilson v, Jun 16 2006 *)
Accumulate[Table[Count[Divisors[n],?OddQ],{n,0,70}]] (* _Harvey P. Dale, Nov 26 2023 *)

PARI

a(n)=local(c);c=0;for(i=1,n,c+=sumdiv(i,X,X%2));c

PARI

for (n=0, 1000, s=n; d=3; while (n>=d, s+=n\d; d+=2); write("b060831.txt", n, " ", s); ) \\ Harry J. Smith, Jul 12 2009

PARI

a(n)=my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2 \\ Charles R Greathouse IV, Jun 18 2015

Python

def A060831(n): return n+sum(n//i for i in range(3,n+1,2)) # Chai Wah Wu, Jul 16 2022

Python

from math import isqrt
def A060831(n): return ((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)+(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(n) = Sum_{i=1..n} A001227(i).

a(n) = a(n-1) + A001227(n).

a(n) = n + floor(n/3) + floor(n/5) + floor(n/7) + floor(n/9) + ...

a(n) = A006218(n) - A006218(floor(n/2)).

a(n) = Sum_{i=1..ceiling(n/2)} A000005(n-i+1). - Wesley Ivan Hurt, Sep 30 2013

a(n) = Sum_{i=floor((n+2)/2)..n} A000005(i). - N. J. A. Sloane, Dec 06 2020, modified by Xiaohan Zhang, Nov 07 2022

G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016

a(n) ~ n*(log(2*n) + 2*gamma - 1) / 2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019

Extensions

Definition simplified by N. J. A. Sloane, Dec 05 2020