A048272 Number of odd divisors of n minus number of even divisors of n.

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

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

An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013 [Supported by a graph. - Vaclav Kotesovec, Mar 01 2023]

From Keith F. Lynch, Jan 20 2024: (Start)

a(n) takes every possible integer value, positive, negative, and zero. Proof: For all nonnegative integers k, a(3^k) = 1+k, a(2^k) = 1-k.

a(n) takes every possible integer value except 1 and -1 infinitely many times. Proof: a(o^(k-1)) = k and a(4*o^(k-1)) = -k for all positive integers k and odd primes o, of which there are infinitely many. a(n) = 0 iff n = 2 (mod 4). a(n) = 1 iff n = 1. a(n) = -1 iff n = 4.

a(n) takes prime value p only for n = o^(p-1), where o is any odd prime.

Terms have a simple pattern that repeats with a period of 4: Positive, zero, positive, negative.

(End)

References

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

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 97, 7(ii).

Links

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

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 and p. 8).

Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n, for n = 1..1000000

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.

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).

Index entries for sequences mentioned by Glaisher.

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_{i>=2} (1+p_i).

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|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^(2*k)) / (1 - x^(2*k)) [Ramanujan]. - Michael Somos, Jul 22 2006

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

From Reinhard Zumkeller, Jan 21 2012: (Start)

a(n) = A001227(n) - A183063(n).

a(A008586(n)) < 0; a(A005843(a(n)) <= 0; a(A016825(n)) = 0; a(A042968(n)) >= 0; a(A005408(n)) > 0. (End)

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

a(2*n - 1) = A099774(n). - Michael Somos, Aug 12 2017

From Paul D. Hanna, Aug 10 2019: (Start)

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) - x^k)^(n-k) = Sum_{n>=0} a(n)*x^(2*n).

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) + x^k)^(n-k) * (-1)^k = Sum_{n>=0} a(n)*x^(2*n). (End)

a(n) = 2*A000005(2n) - 3*A000005(n). - Ridouane Oudra, Oct 15 2019

Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 2*log(2)-1. - Amiram Eldar, Mar 01 2023

Example

a(20) = -2 because 20 = 2^2*5^1 and (1-2)*(1+1) = -2.
G.f. = x + 2*x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + 3*x^9 + 2*x^11 - 2*x^12 + ...

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
# alternative Maple program:
a:= n-> -add((-1)^d, d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018

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 *)
a[n]:=Sum[-(-1)^d, {d, Divisors[n]}] (* Steven Foster Clark, May 04 2018 *)
f[p_, e_] := If[p == 2, 1 - e, 1 + e]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 09 2022 *)

Prog

(PARI) {a(n) = if( n<1, 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)));
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 */
(PARI) {a(n) = my(d = n -> if(frac(n), 0, numdiv(n))); if( n<1, 0, if( n%4, 1, -1) * (d(n) - 2*d(n/2) + 2*d(n/4)))}; /* Michael Somos, Aug 11 2017 */
(Haskell)
a048272 n = a001227 n - a183063 n  -- Reinhard Zumkeller, Jan 21 2012
(Magma) [&+[(-1)^(d+1):d in Divisors(n)] :n in [1..95] ]; // Marius A. Burtea, Aug 10 2019

Crossrefs

Cf. A048298. A diagonal of A060184.

First differences of A059851.

Cf. A001227, A035218, A094572, A099774, A112329, A183063, A000005.

Indices of records: A053624 (highs), A369151 (lows).

Sequence in context: A239703 A029338 A240883 * A112329 A335884 A325033

Adjacent sequences: A048269 A048270 A048271 * A048273 A048274 A048275

Keyword

easy,sign,nice,mult

Author

Adam Kertesz

Extensions

New definition from Vladeta Jovovic, Jan 27 2002

Status

approved