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 A048272 Number of odd divisors of n minus number of even divisors of n. 51
 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. 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). 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). 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 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 a(2*n - 1) = A099774(n). - Michael Somos, Aug 12 2017 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 *) 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 CROSSREFS Cf. A048298. A diagonal of A060184. First differences of A059851. Cf. A001227, A035218, A094572, A099774, 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 EXTENSIONS New definition from Vladeta Jovovic, Jan 27 2002 STATUS approved

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Last modified December 11 18:03 EST 2018. Contains 318049 sequences. (Running on oeis4.)