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A195095 G.f.: Sum_{n>=1} -moebius(2*n)*x^n/(1 - 2*x^n). 1
1, 2, 3, 8, 15, 30, 63, 128, 252, 510, 1023, 2040, 4095, 8190, 16365, 32768, 65535, 131040, 262143, 524280, 1048509, 2097150, 4194303, 8388480, 16777200, 33554430, 67108608, 134217720, 268435455, 536870370, 1073741823, 2147483648, 4294966269, 8589934590 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare the g.f. of this sequence with the following identity: Sum_{n>=1} -moebius(2*n)*x^n/(1-x^n) = Sum_{n>=0} x^(2^n).

Conjecture: Mobius transform of A127804. - R. J. Mathar, Sep 14 2011

a(n) = n*A000048(n), where A000048(n) = number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged. - Paul D. Hanna, Dec 21 2016

LINKS

Table of n, a(n) for n=1..34.

FORMULA

a(2^n) = 2^(2^n - 1).

a(p) = 2^(p-1) for odd prime p.

a(n) = floor(2^(n-1)/n)*n  unless n=3k, k>4. - M. F. Hasler, Sep 08 2011

G.f.: Sum_{n>=1} moebius(2*n-1)*x^(2*n-1)/(1 - 2*x^(2*n-1)). - Mamuka Jibladze, Dec 04 2016

a(n) = Sum_{odd d|n} moebius(d) * 2^(n/d-1), where moebius(n) = A008683(n). - Paul D. Hanna, Dec 21 2016

EXAMPLE

G.f. = x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 30*x^6 + 63*x^7 + 128*x^8 + ...

PROG

(PARI) {a(n)=polcoeff(sum(m=1, n, -moebius(2*m)*x^m/(1-2*x^m+x*O(x^n))^1), n)}

for(n=1, 30, print1(a(n), ", "))

(PARI) {a(n) =  sumdiv(n, d, if(d%2==1, moebius(d) * 2^(n/d-1) ) )}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A011946 (differs from a(15) on). - M. F. Hasler, Sep 08 2011

Cf. A127804, A000048, A008683.

Sequence in context: A099920 A128022 A011946 * A166920 A242510 A080206

Adjacent sequences:  A195092 A195093 A195094 * A195096 A195097 A195098

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 08 2011

STATUS

approved

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Last modified September 15 12:38 EDT 2019. Contains 327078 sequences. (Running on oeis4.)