login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048272 Number of odd divisors of n minus number of even divisors of n. 31
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

abs(a(n))= 1/2* number of pairs (i,j) satisfying n=i^2-j^2 and -n<=i,j<=n - Benoit Cloitre, Jun 14 2003

As A001227(n) is the number of ways to write n as the difference of 3-gonal numbers, a(n) describes the number of ways to write n as the difference of e-gonal numbers for e in {0,1,4,8}. If pe(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 4*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=1, 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e in {0,4} and for a(n) itself is a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=8. (Same for e=-1 see A035218.) - Volker Schmitt (clamsi(AT)gmx.net), Nov 09 2004

a(A008586(n)) < 0; a(A005843(a(n)) <= 0; a(A016825(n)) = 0; a(A042968(n)) >= 0; a(A005408(n)) > 0. [Reinhard Zumkeller, Jan 21 2012]

An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), first formula.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_{o-e}(n).

FORMULA

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n) = Sum_{n >= 1} (-1)^(n-1)*x^n/(1-x^n). Expand Sum 1/(1+x^n) in powers of 1/x.

If n=2^p1*3^p2*5^p3*7^p4*11^p5... a(n)=(1-p1)*product(1+pi), i=2, 3, 4, ...

Multiplicative with a(2^e)=1-e and a(p^e)=1+e if p > 2. - Vladeta Jovovic, Jan 27 2002

a(n) = (-1)*sum(d dividing n, (-1)^d) - Benoit Cloitre, May 12 2003

Moebius transform is period 2 sequence [ 1, -1, ...]. - Michael Somos, Jul 22 2006

G.f.: Sum_{k>0} -(-1)^k x^(k^2) (1+x^(2k))/(1-x^(2k)). - Michael Somos, Jul 22 2006

Equals A051731 * [1, -1, 1, -1, 1,...]. - Gary W. Adamson, Nov 07 2007

a(n) = A001227(n) - A183063(n). [Reinhard Zumkeller, Jan 21 2012]

a(n) = sum_{k=0..n} A081362(k)*A015723(n-k). - Mircea Merca, Feb 26 2014

abs(a(n)) = A112329(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014

From Peter Bala, Jan 07 2015: (Start)

Logarithmic g.f.: log( product {n >= 1} (1 + x^n)^(1/n) ) = sum {n >= 1} a(n)*x^n/n.

a(n) = A001227(n) - A183063(n). By considering the logarithmic generating functions of these three sequences we obtain the identity

( product {n >= 0} (1 - x^(2*n+1))^(1/(2*n+1)) )^2 = product {n >= 1} ( (1 - x^n)/(1 + x^n) )^(1/n). (End)

Dirichlet g.f.: zeta(s)*eta(s) = zeta(s)^2*(1-2^(-s+1)). - Ralf Stephan, Mar 27 2015

EXAMPLE

a(20)=-2 because 20=2^2*5^1 and (1-2)*(1+1)=-2

MAPLE

add(x^n/(1+x^n), n=1..60): series(%, x, 59);

A048272 := proc(n)

    local a;

    a := 1 ;

    for pfac in ifactors(n)[2] do

        if pfac[1] = 2 then

            a := a*(1-pfac[2]) ;

        else

            a := a*(pfac[2]+1) ;

        end if;

    end do:

    a ;

end proc: # Schmitt, sign corrected R. J. Mathar, Jun 18 2016

MATHEMATICA

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)

dif[n_]:=Module[{divs=Divisors[n]}, Count[divs, _?OddQ]-Count[ divs, _?EvenQ]]; Array[dif, 100] (* Harvey P. Dale, Aug 21 2011 *)

PROG

(PARI) a(n)=if(n>0, -sumdiv(n, d, (-1)^d)) /* Michael Somos, Jul 22 2006 */

(PARI)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j); \\ log case

s=-log(prod(j=1, N, (1+x^j)^(1/j))); \\ A048272 Number of odd divisors of n minus number of even divisors of n.

s=serconvol(s, c)

v=Vec(s) \\ Joerg Arndt, May 03 2008

(PARI) a(n)=my(o=valuation(n, 2), f=factor(n>>o)[, 2]); (1-o)*prod(i=1, #f, f[i]+1) \\ Charles R Greathouse IV, Feb 10 2013

(PARI) a(n)=direuler(p=1, n, if(p==2, (1-2*X)/(1-X)^2, 1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

(Haskell)

a048272 n = a001227 n - a183063 n  -- Reinhard Zumkeller, Jan 21 2012

CROSSREFS

Cf. A048298. A diagonal of A060184.

First differences of A059851.

Cf. A001227, A035218, A094572, A112329, A183063.

Sequence in context: A239703 A029338 A240883 * A112329 A117448 A093321

Adjacent sequences:  A048269 A048270 A048271 * A048273 A048274 A048275

KEYWORD

easy,sign,nice,mult

AUTHOR

Adam Kertesz (adamkertesz(AT)worldnet.att.net)

EXTENSIONS

New definition from Vladeta Jovovic, Jan 27 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 30 00:42 EDT 2017. Contains 287304 sequences.