A168508 -id:A168508 - cp's OEIS frontend

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

Henry Bottomley, May 01 2001

nonn

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

			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)

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

Zero together with the partial sums of A001227.

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

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

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 *)

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

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

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

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

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

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

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

A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1

Offset: 0

Ralf Stephan, Dec 19 2004

The definition is that of a linear sequence. Equivalently, define a (0,1) infinite lower triangular matrix T(n,k) (0 <= k <= n) by T(n,k) = 1 if k >= n/2, 0 otherwise, and read it by rows. The triangle T begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
...  The matrix T is used in A168508. [Comment revised by N. J. A. Sloane, Dec 05 2020]
Also, square array A read by antidiagonals upwards: A(n,k) = 1 if k >= n, 0 otherwise.
For n >= 1, T(n,k) = number of partitions of n into k parts of sizes 1 or 2. - Nicolae Boicu, Aug 23 2018
T(n, k) is the number of ways to distribute n balls to k unlabeled urns in such a way that no urn receives more than one ball (see Beeler). - Stefano Spezia, Jun 16 2023

			The array A (on the left) and the triangle T of its antidiagonals (on the right):
  1 1 1 1 1 1 1 1 1 ......... 1
  0 1 1 1 1 1 1 1 1 ........ 0 1
  0 0 1 1 1 1 1 1 1 ....... 0 1 1
  0 0 0 1 1 1 1 1 1 ...... 0 0 1 1
  0 0 0 0 1 1 1 1 1 ..... 0 0 1 1 1
  0 0 0 0 0 1 1 1 1 .... 0 0 0 1 1 1
  0 0 0 0 0 0 1 1 1 ... 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 1 1 .. 0 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 0 1 . 0 0 0 0 1 1 1 1 1

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Proposition 4.2.1 at p. 98.

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Row sums of T (and antidiagonal sums of A) are A008619.
Cf. A079813, A168508.
Cf. A103128, A003056.

rows = 15; A = Array[If[#1 <= #2, 1, 0]&, {rows, rows}]; Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, May 04 2017 *)

from math import isqrt
def A101688(n): return isqrt((m:=n<<1)+1)-(isqrt((m<<2)+8)+1>>1)+1 # Chai Wah Wu, Feb 10 2023

G.f.: 1/((1 - x*y)*(1 - y)).
G.f. of k-th row of the array: x^(k-1)/(1 - x).
T(n, k) = 1 if binomial(k, n-k) > 0, otherwise 0. - Paul Barry, Aug 23 2005
From Boris Putievskiy, Jan 09 2013: (Start)
a(n) = floor((2*A002260(n)+1)/A003056(n)+3).
a(n) = floor((2*n-t*(t+1)+1)/(t+3)), where
t = floor((-1+sqrt(8*n-7))/2). (End)
a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n+1) - 1/2). - Ridouane Oudra, Jul 16 2020
a(n) = A103128(n+1) - A003056(n). - Ridouane Oudra, Apr 09 2022
E.g.f. of k-th column of the array: exp(x)*Gamma(1+k, x)/k!. - Stefano Spezia, Jun 16 2023

Edited by N. J. A. Sloane, Dec 05 2020

A168509 Triangle read by rows, A051731 * A101688.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 27 2009

Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10,...).

A168508 = A101688 * A051731

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 0, 1, 1, 1;
1, 2, 1, 1, 1, 1,
1, 0, 0, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 0, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1;
1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A051731, A101688, A079247, A168508

Triangle read by rows, inverse Mobius transform of A101688; where A051731 = the

inverse Mobius transform operator.

cp's OEIS Frontend

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

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A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.

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A168509 Triangle read by rows, A051731 * A101688.

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