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A048272
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Number of odd divisors of n minus number of even divisors of n.
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28
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| abs(a(n))= 1/2* number of pairs (i,j) satisfying n=i^2-j^2 and -n<=i,j<=n - Benoit Cloitre (benoit7848c(AT)orange.fr), 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]
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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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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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 (vladeta(AT)eunet.rs), Jan 27 2002
a(n)=(-1)*sum(d dividing n, (-1)^d) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (qntmpkt(AT)yahoo.com), Nov 07 2007
a(n) = A001227(n) - A183063(n). [Reinhard Zumkeller, Jan 21 2012]
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EXAMPLE
| a(20)=-2 because 20=2^2*5^1 and (1-2)*(1+1)=-2
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MAPLE
| add(x^n/(1+x^n), n=1..60): series(%, x, 59);
res:=1; ifac:=op(ifactors(n))[2]; for pfac in ifac do; if pfac[1]=2 then res:=res*(pfac[2]-1); else res:=res*(pfac[2]+1); fi; od; a(n):=res; (Schmitt)
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MATHEMATICA
| Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Sep 20 2005)
dif[n_]:=Module[{divs=Divisors[n]}, Count[divs, _?OddQ]-Count[ divs, _?EvenQ]]; Array[dif, 100] (* From Harvey P. Dale, Aug 21 2011 *)
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PROG
| (PARI) a(n)=if(n>0, -sumdiv(n, d, (-1)^d)) /* Michael Somos Jul 22 2006 */
(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
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)
(Haskell)
a048272 n = a001227 n - a183063 n -- Reinhard Zumkeller, Jan 21 2012
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CROSSREFS
| Cf. A048298. A diagonal of A060184.
First differences of A059851.
Cf. A001227, A035218, A112329.
Sequence in context: A118207 A055378 A029338 * A112329 A117448 A093321
Adjacent sequences: A048269 A048270 A048271 * A048273 A048274 A048275
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KEYWORD
| easy,sign,nice,mult
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AUTHOR
| Adam Kertesz (adamkertesz(AT)worldnet.att.net)
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EXTENSIONS
| New definition from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2002
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