A183063 -id:A183063 - cp's OEIS frontend

A066898 Total number of even parts in all partitions of n.

0, 1, 1, 4, 5, 11, 15, 28, 38, 62, 85, 131, 177, 258, 346, 489, 648, 890, 1168, 1572, 2042, 2699, 3475, 4532, 5783, 7446, 9430, 12017, 15106, 19073, 23815, 29827, 37011, 46012, 56765, 70116, 86033, 105627, 128962, 157476, 191359, 232499, 281286, 340180, 409871

Offset: 1

Naohiro Nomoto, Jan 24 2002

Also sum of all even-indexed parts minus the sum of all odd-indexed parts, except the largest parts, of all partitions of n (cf. A206563). - Omar E. Pol, Feb 14 2012
From Omar E. Pol, Apr 06 2023: (Start)
Convolution of A000041 and A183063.
Convolution of A002865 and A362059.
a(n) is also the total number of even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)

			a(5) = 5 because in all the partitions of 5, namely [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], we have a total of 0+1+1+0+2+1+0=5 even parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

Cf. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.

Column 2 of A206563.

a066898 = p 0 1 where
   p e _             0 = e
   p e k m | m < k     = 0
           | otherwise = p (e + 1 - mod k 2) k (m - k) + p e (k + 1) m
-- Reinhard Zumkeller, Mar 09 2012

a066898 = length . filter even . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:=sum(x^(2*j)/(1-x^(2*j)),j=1..60)/product((1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50); # Emeric Deutsch, Feb 17 2006
A066898 := proc(n)
    add(numtheory[tau](k)*combinat[numbpart](n-2*k),k=1..n/2) ;
end proc: # R. J. Mathar, Jun 18 2016

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
(* Clark Kimberling, Mar 08 2012 *)
a[n_] := Sum[DivisorSigma[0, k] PartitionsP[n - 2k], {k, 1, n/2}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 31 2016, after Vladeta Jovovic *)

a(n) = Sum_{k=1..floor(n/2)} tau(k)*numbpart(n-2*k). - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k=0..floor(n/2)} k*A116482(n,k). - Emeric Deutsch, Feb 17 2006
G.f.: (Sum_{j>=1} x^(2*j)/(1-x^(2*j)))/(Product_{j>=1} (1-x^j)). - Emeric Deutsch, Feb 17 2006
a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). - Reinhard Zumkeller, Mar 09 2012
a(n) = (A006128(n) - A209423(n))/2. - Vaclav Kotesovec, May 25 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(3*n/(2*Pi^2))) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

More terms from Vladeta Jovovic, Jan 26 2002

A171977 a(n) = 2^(k+1) where 2^k is the highest power of 2 dividing n.

Original entry on oeis.org

2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 64, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2, 32, 2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2

Offset: 1

Paul Curtz, Nov 19 2010

When read as a triangle in which the n-th row has 2^n terms, every row is the last half of the next one. All the terms are powers of 2. First column = 2*A000079.
The original definition was: a(n) = (A000265(2n+1) - 1) / A000265(2n).
a(n) seems to be the denominator of Euler(2*n+1,1) but I have no proof of this.
a(n) is also gcd[C(2n,1), C(2n,3), ..., C(2n,2n-1)]. - Franz Vrabec, Oct 22 2012
a(n) is also the ratio r(2n) = s2(2n)/s1(2n) where s1(2n) is the sum of the odd unitary divisors of 2n and s2(2n) is the sum of the even unitary divisors of 2n. - Michel Lagneau, Dec 19 2013
a(n) or a(n)/2 = A006519(n) is known as the Steinhaus sequence in probability theory, proposed as a sequence of asymptotically fair premiums for the St. Petersburg game. - Peter Kern, Aug 28 2015
After the all-1's sequence this is the next sequence in lexicographical order such that the gap between a(n) and the next occurrence of a(n) is given by a(n). - Scott R. Shannon, Oct 16 2019
First 2^(k-1) - 1 terms are also the areas of the successive rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure of A139250 after 2^k stages, with k >= 2. For example: if k = 5 the areas after 32 stages are [2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2] respectively, the same as the first 15 terms of this sequence. - Omar E. Pol, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383
Sandor Csörgö and Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157--172. MR1321758 (96b:60017). - _Peter Kern_, Aug 28 2015
Roger B. Eggleton, Aviezri S. Fraenkel, and R. Jamie Simpson, Beatty sequences and Langford sequences, Graph theory and combinatorics (Marseille-Luminy, 1990). Discrete Math. 111 (1993), no. 1-3, 165--178. MR1210094 (94a:11018). See Example 2.6. - _N. J. A. Sloane_, Mar 18 2012
Hugo Steinhaus, The so-called Petersburg paradox, Colloq. Math. 2 (1949), 56--58. MR0039937 (12,619e).

Cf. A000079, A000265, A006519, A038712, A139250.

a := proc(n) local k: k:=1: while frac(n/2^k) = 0 do k := k+1 end do: k := k-1: a(n) := 2^(k+1) end: seq(a(n), n=1..63); # Johannes W. Meijer, Nov 04 2012
seq(2^(1 + padic[ordp](n, 2)), n = 1..63); # Peter Luschny, Nov 27 2020

Table[-BitXor[-i,i], {i, 200}] (* Peter Luschny, Jun 01 2011 *)
a[n_] := 2^(IntegerExponent[n, 2] + 1); Array[a, 100] (* Jean-François Alcover, May 09 2017 *)

A171977(n) = 2^(1+valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

def A171977(n): return (n&-n)<<1 # Chai Wah Wu, Jul 13 2022

a(n) = (A000265(2*n+1)-1)/A000265(2*n).
a(n) = -(-n XOR n).  XOR the bitwise operation on the two's complement representation for negative integers. - Peter Luschny, Jun 01 2011
a(n) = A038712(n)+1. - Franz Vrabec, Mar 03 2012
a(n) = 2^A001511(n). - Franz Vrabec, Oct 22 2012
a(n) = A046161(n)/A046161(n-1). - Johannes W. Meijer, Nov 04 2012
a(n) = 2^(1 + (A183063(n)/A001227(n))). - Omar E. Pol, Nov 06 2018
a(n) = 2*A006519(n). - Antti Karttunen, Nov 06 2018

I edited this sequence, based on an email message from the author. - N. J. A. Sloane, Nov 20 2010

Definition simplified by N. J. A. Sloane, Mar 18 2012

A069734 Number of pairs (p,q), 0<=p<=q, such that p+q divides n.

Original entry on oeis.org

1, 3, 3, 6, 4, 9, 5, 11, 8, 12, 7, 19, 8, 15, 14, 20, 10, 24, 11, 26, 18, 21, 13, 37, 17, 24, 22, 33, 16, 42, 17, 37, 26, 30, 26, 53, 20, 33, 30, 52, 22, 54, 23, 47, 42, 39, 25, 71, 30, 51, 38, 54, 28, 66, 38, 67, 42, 48, 31, 94, 32, 51, 55, 70, 44, 78, 35, 68, 50, 78, 37, 108

Offset: 1

Valery A. Liskovets, Apr 07 2002

Also number of orientable coverings of the Klein bottle with 2n lists (orientable m-list coverings exist only for even m).
Equals row sums of triangle A178650. - Gary W. Adamson, May 31 2010
Also number of inequivalent sublattices of index n of the rectangular lattice, that has the p2mm (pmm) symmetry group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

			There are 9 pairs (p,q), 0<=p<=q, such that p+q divides 6: (0,1), (0,2), (0,3), (0,6), (1,1), (1, 2), (1, 5), (2, 4), (3, 3); thus a(6) = 9.
x + 3*x^2 + 3*x^3 + 6*x^4 + 4*x^5 + 9*x^6 + 5*x^7 + 11*x^8 + 8*x^9 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
V. A. Liskovets and A. Mednykh, Number of non-orientable coverings of the Klein bottle
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2]. [From _N. J. A. Sloane_, Feb 23 2009]
Index entries for sequences related to sublattices

Cf. A069733, A069735, A046524, A178650.
Cf. A000203, A145391-A145394, A003051.
Cf. A304182, A304183, A008619, A007503, A183063.

with(numtheory): a := n -> (sigma(n) + tau(n) + `if`(irem(n,2) = 1, 0, tau(n/2)))/2: seq(a(n), n=1..72); # Peter Luschny, Jul 20 2019

a[n_] := (DivisorSigma[1, n] + DivisorSigma[0, n] + If[OddQ[n], 0, DivisorSigma[0, n/2]])/2;
Array[a, 72] (* Jean-François Alcover, Aug 27 2019, from Maple *)

{a(n) = if( n<1, 0, sum( k=1, n, sum( j=0, k, n%(j+k) == 0)))} /* Michael Somos, Mar 24 2012 */

a(n) = A046524(2n) - A069733(2n).
Inverse Moebius transform of: 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... G.f.: Sum_{n>0} x^n*(1+x^n-x^(2*n))/(1-x^(2*n))/(1-x^n). - Vladeta Jovovic, Feb 03 2003
a(n) = (A000203(n) + A069735(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A304182(n/m^2) + A304183(n/m^2) = A069735(n) + Sum_{ m: m^2|n } A304183(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008619(d) = Sum_{ d|n } (1 + floor(d/2)). - Andrey Zabolotskiy, Jul 20 2019
a(n) = (A007503(n) + A183063(n))/2. - Peter Luschny, Jul 20 2019

New description from Vladeta Jovovic, Feb 03 2003

A112329 Number of divisors of n if n odd, number of divisors of n/4 if n divisible by 4, otherwise 0.

Original entry on oeis.org

1, 0, 2, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 2, 0, 2, 2, 4, 0, 2, 4, 3, 0, 4, 2, 2, 0, 2, 4, 4, 0, 4, 3, 2, 0, 4, 4, 2, 0, 2, 2, 6, 0, 2, 6, 3, 0, 4, 2, 2, 0, 4, 4, 4, 0, 2, 4, 2, 0, 6, 5, 4, 0, 2, 2, 4, 0, 2, 6, 2, 0, 6, 2, 4, 0, 2, 6, 5, 0, 2, 4, 4, 0, 4, 4, 2, 0, 4, 2, 4, 0, 4, 8, 2, 0, 6, 3, 2, 0, 2, 4, 8

Offset: 1

Michael Somos, Sep 04 2005

First occurrence of k: 2, 1, 3, 9, 15, 64, 45, 256, 96, 144, 192, 4096, 240, ????, 768, 576, 480, ????, 720, ..., . See A246063. - Robert G. Wilson v, Oct 31 2013
a(n) is the number of pairs (u, v) in NxZ satisfying u^2-v^2=n. See Kühleitner. - Michel Marcus, Jul 30 2017
The g.f. in the form Sum_{k >= 1} x^(k^2) * (1 + x^(2*k))/(1 - x^(2*k)) = Sum_{k >= 1} x^(k^2) * (1 + x^(2*k))/(1 + x^(2*k) - 2*x^(2*k)) == Sum_{k >= 1} x^(k^2) (mod 2). It follows that a(n) is odd iff n = k^2 for some positive integer k. - Peter Bala, Jan 08 2025

			x + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 3*x^9 + 2*x^11 + 2*x^12 + ...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Ray Chandler, Table of n, a(n) for n = 1..10000
M. Kühleitner, An Omega Theorem on Differences of Two Squares, Acta Mathematica Universitatis Comenianae, Vol. 61, 1 (1992) pp. 117-123. See Lemma 1 p. 2.
John Shareshian and Sheila Sundaram, Ramanujan sums and rectangular power sums, arXiv:2305.12007 [math.CO], 2023. Mentions this sequence.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A000005, A001227, A001620, A048272, A078703, A094572, A099774, A183063, A246063.

f:= proc(n) if n::odd then numtheory:-tau(n) elif n mod 4 = 0 then numtheory:-tau(n/4) else 0 fi end proc;
seq(f(i),i=1..100); # Robert Israel, Aug 24 2014

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
Table[If[OddQ[n],DivisorSigma[0,n],If[OddQ[n/2],0,DivisorSigma[0,n/4]]],{n,100} ] (* Ray Chandler, Aug 23 2014 *)

{a(n) = if( n<1, 0, (-1)^n * sumdiv( n, d, (-1)^d))}

{a(n) = if( n<1, 0, if( n%2, numdiv(n), if( n%4, 0, numdiv(n/4))))} /* Michael Somos, Sep 02 2006 */

d(n) = if (denominator(n)==1, numdiv(n), 0);
a(n) = numdiv(n) - 2*d(n/2) + 2*d(n/4); \\ Michel Marcus, Jul 30 2017

Multiplicative with a(2^e) = e-1 if e>0, a(p^e) = 1+e if p>2.
G.f.: Sum_{k>0} x^k / (1 - (-x)^k) = Sum_{k>0} -(-x)^k / (1 + (-x)^k).
Möbius transform is period 4 sequence [ 1, -1, 1, 1, ...].
G.f.: Sum_{k>=1} x^(k^2) * (1+x^(2*k))/(1-x^(2*k)). - Joerg Arndt, Nov 08 2010
a(4*n + 2) = 0. a(n) = -(-1)^n * A048272(n). a(2*n - 1) = A099774(n). a(4*n) = A000005(n). a(4*n + 1) = A000005(4*n + 1). a(4*n - 1) = 2 * A078703(n).
a(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014
Bisection: a(2*k-1) = A000005(2*k-1), a(2*k) = A183063(2*k) - A001227(2*k), k >= 1. See the Hardy reference, p. 142 where a(n) = sigma^*0(n). - _Wolfdieter Lang, Jan 07 2017
a(n) = d(n) - 2*d(n/2) + 2*d(n/4) where d(n) = 0 if n is not an integer. See Kühleitner.
a(n) = Sum_{d|n} [(d mod 2) = (n/d mod 2)], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Mar 21 2022
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s)).
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (2*gamma-1)*n/2, where gamma is Euler's constant (A001620). (End)
a(n) = (-1)^n * Sum_{d|2*n} cos(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A083910 Number of divisors of n that are congruent to 0 modulo 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 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, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A000007, A001227, A010879, A027750, A183063.
Cf. A001620, A002392.
Cf. A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

a083910 = sum . map (a000007 . a010879) . a027750_row
-- Reinhard Zumkeller, Jan 15 2013

ndc10[n_]:=Count[Divisors[n],?(Divisible[#,10]&)]; Array[ndc10,110] (* _Harvey P. Dale, Jan 05 2013 *)
a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

a(n)=if(n%10,0,numdiv(n/10)) \\ Charles R Greathouse IV, Sep 27 2015

a(n) = A000005(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - Franklin T. Adams-Watters, Apr 15 2007
G.f.: Sum_{k>=1} x^(10*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = (2*gamma - 1 - log(10))/10 = -0.214815..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

A136655 Product of odd divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089

Offset: 1

Jonathan Vos Post, Jun 25 2008

Product of rows of triangle A182469. - Reinhard Zumkeller, May 01 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000265, A000593, A007955, A007956, A078701.
Cf. A140210, A140211, A140213, A140214, A140215.
Cf. A125911, A126192.
Cf. A001227, A183063.

a136655 = product . a182469_row  -- Reinhard Zumkeller, May 01 2012

with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008

Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017

from math import isqrt
from sympy import divisor_count
def A136655(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025

a(p) = p if p noncomposite; a(2^n) = 1; a(pq) = p^2 * q^2 when p, q are odd primes.
a(n) = sqrt(n^od(n)/2^ed(n)), where od(n) = number of odd divisors of n = tau(2*n)-tau(n) and ed(n) = number of even divisors of n = 2*tau(n)-tau(2*n). - Vladeta Jovovic, Jun 25 2008
Also a(n) = A007955(A000265(n)). - David Wilson, Jun 26 2008
a(n) = Product_{h == 1 mod 4 and h | n}*Product_{i == 3 mod 4 and i | n}.
a(n) = Product_{j == 1 mod 6 and j | n}*Product_{k == 5 mod 6 and k | n}.
a(n) = A140210(n)*A140211(n). - R. J. Mathar, Jun 27 2008
a(n) = A007955(n) / A125911(n).

More terms from N. J. A. Sloane, Jul 14 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A299480 List of pairs (a,b) where in the n-th pair, a = number of odd divisors of n and b = number of even divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 1, 3, 3, 0, 2, 2, 2, 0, 2, 4, 2, 0, 2, 2, 4, 0, 1, 4, 2, 0, 3, 3, 2, 0, 2, 4, 4, 0, 2, 2, 2, 0, 2, 6, 3, 0, 2, 2, 4, 0, 2, 4, 2, 0, 4, 4, 2, 0, 1, 5, 4, 0, 2, 2, 4, 0, 3, 6, 2, 0, 2, 2, 4, 0, 2, 6, 2, 0, 4, 4, 2, 0, 2, 4, 6, 0, 2, 2, 2, 0, 2, 8, 3, 0, 3, 3, 4, 0, 2, 4

Offset: 1

Omar E. Pol, Feb 10 2018

Also sequence found by reading in the upper part of the diagram of periodic curves for the number of divisors of n (see the first diagram in the Links section). Explanation: the number of curves that emerge from the point (n, 0) to the left hand in the upper part of the diagram equals A001227(n) the number of odd divisors of n. The number of curves that emerge from the same point (n, 0) to the right hand in the upper part of the diagram equals A183063(n) the number of even divisors of n. So the n-th pair is (A001227(n), A183063(n)). Also the total number of curves that emerges from the same point (n, 0) equals A000005(n), the number of divisors of n. Note that at the point (n, 0) the inflection point of the curve that emerges with diameter k represents the divisor n/k.

The second diagram in the links section shows only the upper part from the first diagram.

			Array begins:
n      A001227  A183063
1         1        0
2         1        1
3         2        0
4         1        2
5         2        0
6         2        2
7         2        0
8         1        3
9         3        0
10        2        2
11        2        0
12        2        4
...

Robert Israel, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of periodic curves for the number of divisors of n (for n = 1..16)
Omar E. Pol, Upper part of the diagram of periodic curves for the number of divisors of n (for n = 1..16)

Row sums give A000005.
For another version see A299485.
Cf. A001227, A027750, A048272, A183063.

f := proc (n) local t; t := numtheory:-tau(n/2^padic:-ordp(n, 2)); t, numtheory:-tau(n)-t end proc:
map(f, [$1..100]); # Robert Israel, Feb 11 2018

m = 105; CoefficientList[Sum[(x^(2n-1) + x^(4n))/(1 - x^(4n)), {n, 1, m/2//Ceiling}] + O[x]^m, x] // Rest (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

Pair(a,b) = Pair(A001227(n), A183063(n)).

G.f.: Sum_{n>=1} (x^(2*n-1) + x^(4*n))/(1-x^(4*n)). - Robert Israel, Feb 11 2018

A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

Original entry on oeis.org

1, 3, 2, 5, 2, 6, 2, 7, 3, 6, 2, 10, 2, 6, 4, 9, 2, 9, 2, 10, 4, 6, 2, 14, 3, 6, 4, 10, 2, 12, 2, 11, 4, 6, 4, 15, 2, 6, 4, 14, 2, 12, 2, 10, 6, 6, 2, 18, 3, 9, 4, 10, 2, 12, 4, 14, 4, 6, 2, 20, 2, 6, 6, 13, 4, 12, 2, 10, 4, 12, 2, 21, 2, 6, 6, 10, 4, 12, 2, 18, 5, 6, 2, 20, 4, 6, 4, 14, 2, 18

Offset: 1

Valery A. Liskovets, Apr 07 2002

Dirichlet convolution of A000012 by A040001. - R. J. Mathar, Mar 30 2011

a(n) is the number of full-dimensional lattices with volume n in Z^2 which are symmetric about a coordinate axis (equivalently, about both). - Álvar Ibeas, Mar 19 2021

			x + 3*x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 3*x^9 + 6*x^10 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

Equals row sums of triangle A143110. - Gary W. Adamson, Jul 25 2008

Cf. A000005, A000012, A001227, A001620, A040001, A059841, A069734, A099774, A183063, A304182, A320111.

read("transforms") : nmax := 100 :
L := [1,1,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
with(NumberTheory): seq(tau(n) + `if`(n::odd, 0, tau(n/2)), n=1..100); # Peter Luschny, Mar 19 2021

d[n_] := DivisorSigma[0, n];
a[n_] := If[EvenQ[n], d[n] + d[n/2], d[n]];
Array[a, 100] (* Jean-François Alcover, Aug 27 2019 *)

{a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* Michael Somos, Mar 24 2012 */

Multiplicative with a(2^e)=2e+1 and a(p^e)=e+1 for e>0 and an odd prime p.
a(n) = d(n)+d(n/2) for even n and a(n) = d(n) otherwise where d(n) is the number of divisors of n (A000005).
G.f.: Sum_{k>0} x^k*(1+2*x^k)/(1-x^(2*k)). - Vladeta Jovovic, Dec 16 2002
Dirichlet g.f.: (1+2^(-s))*zeta^2(s) [ Rutherford]. - N. J. A. Sloane, Feb 23 2009
Moebius transform is period 2 sequence [ 1, 2, ...]. - Michael Somos, Mar 24 2012
a(2*n - 1) = A099774(n).
a(n) = Sum_{ m: m^2|n } A304182(n/m^2). - Andrey Zabolotskiy, May 07 2018
Sum_{k=1..n} a(k) ~ 3*n*log(n)/2 + (3*gamma - 3/2 - log(2)/2)*n, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 04 2019
a(n) = 3*tau(n) - tau(2*n). - Ridouane Oudra, Mar 15 2021
a(n) = A320111(n) + (A059841(n)*A000005(n)), i.e. a(n) = A320111(n) if n is odd, and a(n) = A320111(n) + A000005(n) if n is even. - Antti Karttunen, Mar 17 2021
a(n) = A000005(n) + A183063(n) = 2*A000005(n) - A001227(n). - Amiram Eldar, Dec 22 2023

Corrected by T. D. Noe, Nov 13 2006

A299485 List of pairs (a,b) where in the n-th pair, a = number of even divisors of n and b = number of odd divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 3, 1, 0, 3, 2, 2, 0, 2, 4, 2, 0, 2, 2, 2, 0, 4, 4, 1, 0, 2, 3, 3, 0, 2, 4, 2, 0, 4, 2, 2, 0, 2, 6, 2, 0, 3, 2, 2, 0, 4, 4, 2, 0, 2, 4, 4, 0, 2, 5, 1, 0, 4, 2, 2, 0, 4, 6, 3, 0, 2, 2, 2, 0, 4, 6, 2, 0, 2, 4, 4, 0, 2, 4, 2, 0, 6, 2, 2, 0, 2, 8, 2, 0, 3, 3, 3, 0, 4, 4, 2

Offset: 1

Omar E. Pol, Mar 03 2018

Also sequence found by reading in the lower part of the diagram of periodic curves for the number of divisors of n (see the first diagram in the Links section). Explanation: the number of curves that emerge from the point (n, 0) to the left hand in the lower part of the diagram equals A183063(n) the number of even divisors of n. The number of curves that emerge from the same point (n, 0) to the right hand in the lower part of the diagram equals A001227(n) the number of odd divisors of n. So the n-th pair is (A183063(n), A001227(n)). Also the total number of curves that emerges from the same point (n, 0) equals A000005(n), the number of divisors of n. Note that at the point (n, 0) the inflection point of the curve that emerges with diameter k represents the divisor n/k.

The second diagram in the links section shows only the lower part from the first diagram, upside down.

			Array begins:
n      A183063  A001227
1         0        1
2         1        1
3         0        2
4         2        1
5         0        2
6         2        2
7         0        2
8         3        1
9         0        3
10        2        2
11        0        2
12        4        2
...

Another version of A299480.
Row sums give A000005.
Cf. A001227, A027750, A048272, A183063.

Array[{#2, #1 - #2} & @@ {DivisorSigma[0, #], DivisorSum[#, 1 &, EvenQ]} &, 52] // Flatten (* Michael De Vlieger, Mar 04 2018 *)

Pair(a,b) = Pair(A183063(n), A001227(n)).

A129372 Triangle read by rows: T(n,k) = 1 if k divides n and n/k is odd, T(n,k) = 0 otherwise.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 11 2007

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A051731, A115359, A183063.

Cf. A001227 (row sums).

A129372:= func< n,k | (n mod k) eq 0 and (Floor(n/k) mod 2) eq 1 select 1 else 0 >;
[A129372(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Feb 01 2024

A129372[n_, k_]:= If[Mod[n,k]==0 && OddQ[n/k], 1, 0];
Table[A129372[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Feb 01 2024 *)

T(n,k)=if(n%k, 0, n/k%2==1) \\ Andrew Howroyd, Aug 10 2018

def A129372(n,k): return 1 if (n%k)==0 and ((n/k)%2)==1 else 0
flatten([[A129372(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Feb 01 2024

Equals A051731 * A115359.
Sum_{k=1..n} T(n, k) = A001227(n) (row sums).
From G. C. Greubel, Feb 01 2024: (Start)
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A001227(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A183063(n+1). (End)

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

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