A001227 -id:A001227 - 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

A320107 a(n) = A001227(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 4, 2, 1, 2, 2, 4, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 2, 3, 2, 2, 3, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Records 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, ... occur at n = 1, 5, 18, 30, 90, 210, 450, 630, 1890, 3150, 5670, 6930, 20790, 34650, 62370, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001227, A005940, A051064, A055457, A252463, A320106 (Möbius transform).

Cf. also A320111 and A319699, A319700, A319703, A319989.

A001227(n) = numdiv(n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A320107(n) = A001227(A252463(n));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A320107(n) = if(n<=2, 1, my(p=A252463(n)); if(!(p%2), A320107(p), A051064(p)*A320107(p)));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A055457(n) = if(n<1, 0, 1+valuation(n, 5));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A320107(n) = if(n<=2, 1, if(!(n%4), A320107(n/2), if(2==(n%4), A051064(n)*A320107(n/2), if(!(n%3), A320107(A064989(n)), A055457(n)*A320107(A064989(n))))));

a(n) = A001227(A252463(n)).
a(1) = a(2) = 1; for n > 2, a(n) = a(n/2) when n == 0 mod 4, a(n) = A051064(n) * a(n/2) when n == 2 mod 4, a(n) = a(A064989(n)), when n == 3 mod 6, otherwise a(n) = A055457(n) * a(A064989(n)).
For n > 2, let p = A252463(n). If p is even, then a(n) = a(p), if p is odd, then a(n) = A051064(p) * a(p).

A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 3, 4, 1, 2, 2, 4, 2, 4, 1, 4, 1, 2, 2, 4, 2, 4, 1, 2, 4, 4, 1, 2, 1, 2, 2, 8, 1, 2, 2, 3, 4, 2, 1, 2, 3, 2, 4, 6, 1, 2, 1, 4, 2, 6, 2, 4, 1, 4, 2, 2, 1, 4, 1, 4, 2, 4, 2, 4, 1, 2, 4, 4, 1, 6, 4, 8, 8, 8, 1, 6, 3, 4, 6, 12, 4, 4, 1, 3, 4, 4, 1, 6, 1, 2, 4

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4473
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001227, A156552, A246277, A322813, A322993, A324105, A324115, A324118.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A322993(n) = A156552(2*A246277(n));
A324117(n) = if(1==n, 0, numdiv(A322993(n)));

a(1) = 0; for n > 1, a(n) = A000005(A322993(n)) = A000005(A156552(2*A246277(n))) = A324105(2*A246277(n)).

A322813 a(n) = A001227(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 4, 2, 3, 2, 1, 4, 1, 2, 3, 2, 4, 4, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 5, 6, 3, 2, 1, 4, 4, 2, 3, 2, 1, 4, 1, 2, 3, 2, 4, 4, 1, 2, 3, 6, 1, 4, 1, 2, 6, 2, 5, 4, 1, 2, 3, 2, 1, 4, 4, 2, 3, 2, 1, 4, 5, 2, 3, 2, 4, 2, 1, 8, 3, 6, 1, 4, 1, 2, 6

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A001227, A122111, A322819.

A001227(n) = numdiv(n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322813(n) = A001227(A122111(n));

a(n) = A001227(A122111(n)).

A133698 Triangle, diagonal = A001227 with the rest zeros.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson, Sep 21 2007

Lower triangular part of an infinite matrix with A001227 (number of odd divisors of n) as the main diagonal, and the rest filled with zeros. - Redacted from the original formula given by the author. - Antti Karttunen, Jan 18 2025

			First few rows of the triangle are:
  1;
  0, 1;
  0, 0, 2
  0, 0, 0, 1;
  0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 3;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;
  ...

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A001227, A130209, A133699.

A000265(n) = (n>>valuation(n,2));
A133698(n) = if(ispolygonal(n,3), numdiv(A000265((sqrtint(1+(n*8))-1)/2)), 0); \\ Antti Karttunen, Jan 18 2025

Offset corrected from 0 to 1 and data section extended to a(105) by Antti Karttunen, Jan 18 2025

A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 3, 4, 6, 6, 3, 9, 3, 6, 9, 5, 3, 12, 3, 9, 9, 6, 3, 12, 6, 6, 10, 9, 3, 18, 3, 6, 9, 6, 9, 18, 3, 6, 9, 12, 3, 18, 3, 9, 18, 6, 3, 15, 6, 12, 9, 9, 3, 20, 9, 12, 9, 6, 3, 27, 3, 6, 18, 7, 9, 18, 3, 9, 9, 18, 3, 24, 3, 6, 18, 9, 9, 18, 3, 15, 15, 6, 3, 27, 9, 6, 9, 12, 3, 36, 9, 9, 9

Offset: 1

Gary W. Adamson, Sep 21 2007

			a(4) = sum of row 4 terms of triangle A133699: (1 + 1 + 0 + 1) = (1, 1, 0, 1) dot (1, 1, 2, 1), where A001227 = (1, 1, 2, 1, 2, 2, 2, 1, 3, ...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A001227, A051731, A133699.

Cf. A001620, A082633.

f[p_, e_] := (e+1)*(e+2)/2; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)

A133700(n) = sumdiv(n,d,numdiv(d>>valuation(d,2))); \\ Antti Karttunen, Sep 27 2018

Inverse Möbius transform of A001227, the number of odd divisors of n. Row sums of triangle A133699.
Dirichlet g.f. (zeta(s))^3*(1-1/2^s). - R. J. Mathar, Feb 07 2011
a(n) = Sum_{d|n} A001227(d). - Antti Karttunen, Sep 27 2018
Sum_{k=1..n} a(k) ~ n/4 * (log(n)^2 + (6*g - 2 + 2*log(2))*log(n) + 2 + 6*g^2 - log(2)^2 - 2*log(2) + 6*g*(log(2) - 1) - 6*sg1), where g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Feb 02 2019
G.f.: Sum_{k>=1} tau(k)*x^k/(1 - x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 13 2019
Multiplicative with a(2^e) = e+1, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 28 2023

More terms from R. J. Mathar, Jan 19 2009

Second, equivalent formula added to the definition by Antti Karttunen, Sep 27 2018

A205965 a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.

Original entry on oeis.org

1, 1, 4, 3, 10, 16, 26, 21, 102, 110, 178, 288, 466, 754, 2440, 987, 3194, 7752, 8362, 13530, 43784, 35422, 57314, 92736, 225075, 242786, 785672, 635622, 1028458, 3328160, 2692538, 2178309, 14098312, 11405774, 36909860, 44791056, 48315634, 78176338, 252983944

Offset: 1

Paul D. Hanna, Feb 03 2012

nonn

Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).

			G.f.: A(x) = x + x^2 + 4*x^3 + 3*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 21*x^8 +...
where A(x) = 1*1*x + 1*1*x^2 + 2*2*x^3 + 3*1*x^4 + 5*2*x^5 + 8*2*x^6 + 13*2*x^7 + 21*1*x^8 +...+ Fibonacci(n)*A001227(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1*x/(1-x-x^2) + 2*x^3/(1-4*x^3-x^6) + 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) + 34*x^9/(1-76*x^9-x^18) + 89*x^11/(1-199*x^11-x^22) +...
which involves odd-indexed Fibonacci and Lucas numbers.

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A001227, A205964, A205966, A203847, A000204 (Lucas).

Cf. A209445 (Pell variant).

A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(2*m-1)*x^(2*m-1)/(1-Lucas(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

a(n) = fibonacci(n)*sumdiv(n, d, d%2); \\ Michel Marcus, Jul 18 2018

G.f.: Sum_{n>=1} Fibonacci(2*n-1)*x^(2*n-1)/(1 - Lucas(2*n-1)*x^(2*n-1)-x^(4*n-2)).

A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

Original entry on oeis.org

1, 1, 3, 19, 97, 801, 7411, 73123, 821409, 10977697, 151612291, 2286137811, 38308830913, 669163118209, 12649211055027, 257559356068771, 5432325991339201, 121949878889492673, 2907330680764076419, 71860237654425159187, 1871308081194213959841

Offset: 0

Seiichi Manyama, Oct 30 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..441

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294394 (k=1), A294395 (k=2).

Cf. A001227, A206303.

a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSum[k, Mod[#, 2] &]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^(2*k-1)/(1 - x^(2*k-1))). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A294459 E.g.f.: exp(-Sum_{n>=1} A001227(n) * x^n).

Original entry on oeis.org

1, -1, -1, -7, 25, -41, 631, 881, 98897, -609265, 3798991, -41799671, 914146729, -15008576857, 16469525255, -5181463756351, 79515495724321, -1220435382764129, 12608713897126687, -449855614172366695, 10437031873016276921, -231918657853281955081

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..446

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294460 (k=1), A294461 (k=2).

Cf. A018804.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} 1 / (1 + x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j). - Ilya Gutkovskiy, Aug 17 2021

A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (16384 terms)

Cf. A001227.
Cf. A318454 (gives the denominators).
Differs from A318313 for the first time at n=81, where a(81) = 1, while A318313(81) = 3.

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318453(n) = numerator(v318453_54[n]);
A318454(n) = denominator(v318453_54[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318453(k) / A318454(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

cp's OEIS Frontend

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

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A320107 a(n) = A001227(A252463(n)).

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A324117 Number of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A001227(A156552(n)) = A000005(A322993(n)).

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A322813 a(n) = A001227(A122111(n)).

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A133698 Triangle, diagonal = A001227 with the rest zeros.

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A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

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A205965 a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.

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