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A067856 Sum_{n >= 1} a(n)/n^s = 1/(Sum_{n >= 1} (-1)^(n+1)/n^s). 12
1, 1, -1, 2, -1, -1, -1, 4, 0, -1, -1, -2, -1, -1, 1, 8, -1, 0, -1, -2, 1, -1, -1, -4, 0, -1, 0, -2, -1, 1, -1, 16, 1, -1, 1, 0, -1, -1, 1, -4, -1, 1, -1, -2, 0, -1, -1, -8, 0, 0, 1, -2, -1, 0, 1, -4, 1, -1, -1, 2, -1, -1, 0, 32, 1, 1, -1, -2, 1, 1, -1, 0, -1, -1, 0, -2, 1, 1, -1, -8, 0, -1, -1, 2, 1, -1, 1, -4, -1, 0, 1, -2, 1, -1, 1, -16, -1, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Dirichlet inverse of A062157. - R. J. Mathar, Jul 15 2010

The first 31 terms equal the values of the Ramanujan sum c_n(8) -- see for example A085906 -- but a(32) <> c_{32}(8). - R. J. Mathar, Apr 02 2011

From  Peter Bala, Mar 12 2019: (Start)

Let Mu(n) = (-1)^(n+1)*a(n), an analog of the Möbius function mu(n). Then for arithmetic functions f(n) and g(n) we have the following analog of the Möbius inversion formula: f(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*g(d) iff g(n) = Sum_{d divides n} (-1)^((1+d)*(1+n/d))*Mu(n/d)*f(d).

Each of the following two equations implies the other: F(x) = Sum_{n >= 1} (-1)^(n+1)*G(n*x); G(x) = Sum_{n >= 1} a(n)*F(n*x). See G. Pólya and G. Szegő, Part V111, Chap. 1, Nos. 66-68.2. (End)

Let D(n) denote the set of partitions of n into distinct parts. Then Sum_{parts k in D(n)} a(k) = |D(n-1)| = A000009(n-1). For example, D(6) = {6, 1 + 5, 2 + 4, 1 + 2 + 3} and the sum a(6) + (a(1) + a(5)) + (a(2) + a(4)) + (a(1) + a(2) + a(3)) = 3 = |D(5)|. - Peter Bala, Mar 14 2019

REFERENCES

G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

P. Bala, A signed Dirichlet product of arithmetical functions.

Franz Rogel, Transformationen arithmetischer Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article LI (1897), Prague (31 pages); see pp. 10-11 and especially Eqs. (21) - (24). [There are some obvious typos there; especially Eq. (24) should become Sum_{t|v} (-1)^(v/t) * c(t) = 0 for v > 1, which is the equation a(n) = Sum{k|n, 1<k} (-1)^k a(n/k), for n >= 2, in the FORMULA section below. - Petros Hadjicostas, Jul 21 2019]

FORMULA

a(1) = 1, a(n) = Sum{k|n, 1<k} (-1)^k a(n/k), for n >= 2; sum over divisors, k, of n, where k > 1. If n is odd, a(n) = mu(n), mu() is the Moebius function. If n is even, a(n) = mu(m) 2^(k-1), where n = m*2^k, m is odd integer, k = integer.

Sum_{n>0} a(n)*x^n/(1+x^n) = x. Moebius transform of A048298. Multiplicative with a(2^e) = 2^(e-1), a(p) = -1 for p>2, a(p^e) = 0 for p>2 and e>1. - Vladeta Jovovic, Jan 02 2003

Sum_{n>0} a(n)*log(1+x^n)/n = x. - Paul D. Hanna, May 06 2003

a(n)=0 if and only if n is divisible by square of odd prime (A038838). - Michael Somos, Aug 22 2006

1 = Sum_(k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 2*floor(x/2). - Benoit Cloitre, Nov 11 2010

Dirichlet g.f. 1/( zeta(s) * (1-2^(1-s)) ). - R. J. Mathar, Apr 02 2011

From Peter Bala, Mar 13 2019: (Start)

Sum_{n >= 1} a(n)*x^n/(1 + x^n) = x

Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 2*x^2 + 4*x^4 + 8*x^8 + 16*x^16 + ...

Sum_{n >= 1} a(n)*x^n/(1 + (-x)^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...)

Sum_{n >= 1} a(n)*x^n/(1 - (-x)^n) = x + 2*(x^4 + 3*x^8 + 7*x^16 + 15*x^32 + ...). (End)

G.f. A(x) satisfies: A(x) = x + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... - Ilya Gutkovskiy, May 11 2019

MATHEMATICA

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];

a /@ Range[1, 100] (* Jean-François Alcover, Oct 01 2019 *)

PROG

(PARI) {a(n)=local(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1))} /* Michael Somos, Aug 22 2006 */

(PARI) A067856(n) =  { my(f=factor(n)); for(i=1, #f~, if(2==f[i, 1], f[i, 2]--, if(f[i, 2]>1, f[i, 1]=0, f[i, 1]=-1))); factorback(f); }; \\ Antti Karttunen, Sep 27 2018, after Vladeta Jovovic_'s multiplicative formula.

CROSSREFS

Cf. A038712, A062157, A321088 (Euler transform), A048298 (inverse Mobius transform), A321588.

Sequence in context: A054772 A294616 A085384 * A160467 A122374 A261960

Adjacent sequences:  A067853 A067854 A067855 * A067857 A067858 A067859

KEYWORD

sign,mult

AUTHOR

Leroy Quet, Feb 15 2002

STATUS

approved

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)