A001227 - OEIS

%I #344 Jan 05 2025 19:51:31

%S 1,1,2,1,2,2,2,1,3,2,2,2,2,2,4,1,2,3,2,2,4,2,2,2,3,2,4,2,2,4,2,1,4,2,

%T 4,3,2,2,4,2,2,4,2,2,6,2,2,2,3,3,4,2,2,4,4,2,4,2,2,4,2,2,6,1,4,4,2,2,

%U 4,4,2,3,2,2,6,2,4,4,2,2,5,2,2,4,4,2,4,2,2,6,4,2,4,2,4,2,2,3,6,3,2,4,2,2,8

%N Number of odd divisors of n.

%C Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - _Tom Verhoeff_

%C Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - _Henry Bottomley_, Apr 13 2000

%C This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - _Gus Wiseman_, Oct 04 2022

%C a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

%C Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - _T. D. Noe_, Apr 16 2003

%C a(n) = 1 iff n is a power of 2 (see A000079). - _Lekraj Beedassy_, Apr 12 2005

%C Number of occurrences of n in A049777. - _Philippe Deléham_, Jun 19 2005

%C For n odd, n is prime iff the n-th term of the sequence is 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005

%C Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Mar 07 2006

%C Also the number of factors of the n-th Lucas polynomial. - _T. D. Noe_, Mar 09 2006

%C Lengths of rows of triangle A182469;

%C Denoted by Delta_0(n) in Glaisher 1907. - _Michael Somos_, May 17 2013

%C Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - _Clark Kimberling_, Apr 18 2014

%C Row sums of triangle A247795. - _Reinhard Zumkeller_, Sep 28 2014

%C Row sums of triangle A237048. - _Omar E. Pol_, Oct 24 2014

%C A069288(n) <= a(n). - _Reinhard Zumkeller_, Apr 05 2015

%C A000203, A000593 and this sequence have the same parity: A053866. - _Omar E. Pol_, May 14 2016

%C a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - _Juri-Stepan Gerasimov_, May 23 2016, Jul 15 2016

%C Also the number of odd divisors of n*2^m for m >= 0. - _Juri-Stepan Gerasimov_, Jul 15 2016

%C a(n) is odd iff n is a square or twice a square. - _Juri-Stepan Gerasimov_, Jul 17 2016

%C a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - _Omar E. Pol_, Nov 05 2016

%C a(n) is also the number of partitions of n into an odd number of equal parts. - _Omar E. Pol_, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - _N. J. A. Sloane_, Dec 03 2020]

%C The smallest integer with exactly m odd divisors is A038547(m). - _Bernard Schott_, Nov 21 2021

%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.

%D J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).

%D Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.

%D P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

%H N. J. A. Sloane, <a href="/A001227/b001227.txt">Table of n, a(n) for n = 1..10000</a>

%H K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath107.htm">Partitions into Consecutive Integers</a>.

%H Atli Fannar Franklín, <a href="https://arxiv.org/abs/2410.07467">Pattern avoidance enumerated by inversions</a>, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H A. Heiligenbrunner, <a href="http://ah9.at/ahsummen.htm">Sum of adjacent numbers (in German)</a>.

%H Gerzson Kéri, <a href="http://ac.inf.elte.hu/Vol_053_2022/093_53.pdf">The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas</a>, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).

%H M. A. Nyblom, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-3/nyblom.pdf">On the representation of the integers as a difference of nonconsecutive triangular numbers</a>, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

%H R. C. Read, <a href="/A000684/a000684_1.pdf">Letter to N. J. A. Sloane, Oct. 29, 1976</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialNumber.html">Binomial Number</a> and <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PolygammaFunction.html">q-Polygamma Function</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.

%F Dirichlet g.f.: zeta(s)^2*(1-1/2^s).

%F Comment from _N. J. A. Sloane_, Dec 02 2020: (Start)

%F By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:

%F A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...

%F   = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))

%F   = Sum_{k >= 1} x^k/(1-x^(2*k))

%F   = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].

%F (This incorporates comments from _Vladeta Jovovic_, Oct 16 2002 and  _Michael Somos_, Oct 30 2005.) (End)

%F G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - _Joerg Arndt_, Nov 06 2010

%F a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).

%F Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - _David W. Wilson_, Aug 01 2001

%F a(n) = A000005(A000265(n)). - _Lekraj Beedassy_, Jan 07 2005

%F Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.

%F a(n) = A113414(2*n). - _N. J. A. Sloane_, Jan 24 2006 (corrected Nov 10 2007)

%F a(n) = A001826(n) + A001842(n). - _Reinhard Zumkeller_, Apr 18 2006

%F Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - _Gary W. Adamson_, Apr 15 2007

%F Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - _Gary W. Adamson_, Nov 06 2007

%F a(n) = A000005(n) - A183063(n).

%F a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - _Gary W. Adamson_, Mar 15 2011

%F Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - _R. J. Mathar_, Jun 28 2011

%F a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - _Reinhard Zumkeller_, May 01 2012

%F a(n) = 1 + A069283(n). - _R. J. Mathar_, Jun 18 2015

%F a(A002110(n)/2) = n, n >= 1. - _Altug Alkan_, Sep 29 2015

%F a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - _Juri-Stepan Gerasimov_, Jul 17 2016

%F Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - _Omar E. Pol_, Feb 15 2017

%F a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - _Danny Rorabaugh_, Oct 03 2017

%F L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, Jul 30 2018

%F G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - _Michael Somos_, Jun 01 2019

%F a(n) = A003056(n) - A238005(n). - _Omar E. Pol_, Sep 12 2021

%F Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Nov 27 2022

%F Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - _Amiram Eldar_, Mar 01 2023

%F a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - _Ridouane Oudra_, Apr 13 2023

%e G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...

%e From _Omar E. Pol_, Nov 30 2020: (Start)

%e For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.

%e Illustration of initial terms:

%e                               Diagram

%e    n   a(n)                         _

%e    1     1                        _|1|

%e    2     1                      _|1 _|

%e    3     2                    _|1  |1|

%e    4     1                  _|1   _| |

%e    5     2                _|1    |1 _|

%e    6     2              _|1     _| |1|

%e    7     2            _|1      |1  | |

%e    8     1          _|1       _|  _| |

%e    9     3        _|1        |1  |1 _|

%e   10     2      _|1         _|   | |1|

%e   11     2    _|1          |1   _| | |

%e   12     2   |1            |   |1  | |

%e ...

%e a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

%p for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:

%p A001227 := proc(n) local a,d;

%p     a := 1 ;

%p     for d in ifactors(n)[2] do

%p         if op(1,d) > 2 then

%p             a := a*(op(2,d)+1) ;

%p         end if;

%p     end do:

%p     a ;

%p end proc: # _R. J. Mathar_, Jun 18 2015

%t f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* _Robert G. Wilson v_, Aug 27 2004 *)

%t Table[Total[Mod[Divisors[n], 2]],{n,105}] (* _Zak Seidov_, Apr 16 2010 *)

%t f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* _Robert G. Wilson v_ *)

%t a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* _Michael Somos_, May 17 2013 *)

%t a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* _Michael Somos_, May 17 2013 *)

%t Count[Divisors[#],_?OddQ]&/@Range[110] (* _Harvey P. Dale_, Feb 15 2015 *)

%t (* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)

%t (* cl = current level, cs = current subparts count *)

%t a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]<cl, cl--]]; cs]

%t a001227[105] (* sequence data *) (* _Hartmut F. W. Hoft_, Dec 16 2016 *)

%t a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* _Amiram Eldar_, Jun 12 2022 *)

%o (PARI) {a(n) = sumdiv(n, d, d%2)}; /* _Michael Somos_, Oct 06 2007 */

%o (PARI) {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* _Michael Somos_, Oct 06 2007 */

%o (PARI) a(n)=numdiv(n>>valuation(n,2)) \\ _Charles R Greathouse IV_, Mar 16 2011

%o (PARI) a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ _Charles R Greathouse IV_, May 31 2013

%o (PARI) a(n)=sumdivmult(n,d,d%2) \\ _Charles R Greathouse IV_, Aug 29 2013

%o (Haskell)

%o a001227 = sum . a247795_row

%o -- _Reinhard Zumkeller_, Sep 28 2014, May 01 2012, Jul 25 2011

%o (SageMath)

%o def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])

%o [A001227(n) for n in (1..80)]  # _Peter Luschny_, Feb 01 2012

%o (Magma) [NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // _Vincenzo Librandi_, Jun 02 2019

%o (Python)

%o from functools import reduce

%o from operator import mul

%o from sympy import factorint

%o def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # _Chai Wah Wu_, Mar 08 2021

%Y Cf. A000005, A000079, A000593, A010054 (char. func.), A038547, A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.

%Y Cf. A000203, A001620, A002162, A053866, A060831.

%Y If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.

%Y Dirichlet inverse is A327276.

%K nonn,easy,nice,mult,core

%O 1,3

%A _N. J. A. Sloane_