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A067856 Sum{n=1 to infinity} a(n)/n^s = 1/(sum{n=1 to infinity} (-1)^(n+1)/n^s). 0
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

LINKS

Table of n, a(n) for n=1..100.

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

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 */

CROSSREFS

Cf. A038712.

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 November 18 17:56 EST 2017. Contains 294894 sequences.