A238270 -id:A238270 - cp's OEIS frontend

A238271 Decimal expansion of sum(n>=1, mu(n)/3^n ).

1, 8, 1, 9, 9, 5, 3, 8, 6, 7, 0, 2, 6, 3, 3, 8, 8, 7, 8, 2, 7, 8, 0, 0, 1, 0, 0, 3, 0, 0, 5, 6, 5, 5, 7, 3, 2, 2, 6, 3, 4, 4, 9, 8, 0, 1, 3, 5, 3, 8, 3, 0, 6, 1, 7, 5, 9, 2, 4, 9, 4, 7, 7, 1, 6, 3, 2, 4, 2, 0, 4, 8, 3, 6, 8, 5, 0, 5, 4, 1, 6, 4, 2, 2, 8, 9, 8, 3, 9, 5, 7, 7, 6, 4, 9, 2, 3, 9, 9, 0, 2, 6, 9, 9, 4

Offset: 0

Robert G. Wilson v, Feb 21 2014

Suggested by a posting to SeqFan list by Alonso Del Arte at 15:35 on Aug 09 2012.

			=0.1819953867026338878278001003005655732263449801353830617592494771...

Cf. A008683, A238270, A181673 & A238273.

RealDigits[ Sum[ N[ MoebiusMu[n] 3^-n, 128], {n, 300}], 10, 111][[1]]

A344432 a(n) = Sum_{k=1..n} mu(k) * 2^(n - k).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 7, 13, 26, 52, 105, 209, 418, 835, 1671, 3343, 6686, 13371, 26742, 53483, 106966, 213933, 427867, 855733, 1711466, 3422932, 6845865, 13691730, 27383460, 54766919, 109533837, 219067673, 438135346, 876270693, 1752541387, 3505082775, 7010165550

Offset: 0

Seiichi Manyama, May 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A127513, A238270, A292524, A343425, A344433.

a[n_] := Sum[MoebiusMu[k] * 2^(n-k), {k,1,n}]; Array[a, 40] (* Amiram Eldar, May 19 2021 *)

a(n) = sum(k=1, n, moebius(k)*2^(n-k));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, moebius(k)*x^k)/(1-2*x)))

a(n) = if(n==0, 0, 2*a(n-1)+moebius(n));

G.f.: (Sum_{k>=1} mu(k) * x^k) / (1 - 2*x).
a(n) = 2 * a(n-1) + mu(n) for n > 0.
a(n) ~ A238270 * 2^n. - Vaclav Kotesovec, May 19 2021

A181673 Decimal expansion of Sum_{n>=1} moebius(n)*(1/10)^n.

Original entry on oeis.org

0, 8, 8, 9, 9, 0, 9, 0, 0, 0, 8, 9, 9, 1, 0, 9, 8, 9, 9, 0, 1, 0, 9, 0, 0, 0, 9, 9, 8, 8, 9, 0, 1, 1, 0, 9, 9, 1, 0, 9, 8, 8, 9, 0, 0, 0, 9, 0, 0, 0, 0, 9, 9, 0, 1, 0, 1, 0, 8, 9, 9, 1, 0, 0, 0, 8, 9, 0, 0, 8, 8, 9, 9, 1, 0, 0, 0, 8, 9, 0, 0, 0, 9, 0, 1, 1, 0, 9, 9

Offset: 0

Mats Granvik, Nov 04 2010

			0.088990900089910989901090009988901109910...

Cf. A008683, A238270, A238271, A238273.

RealDigits[ Sum[ N[ MoebiusMu[n] 10^-n, 128], {n, 120}], 10, 111][[1]] (* Robert G. Wilson v, Feb 21 2014 *)

sum(n=1,default(realprecision)+1,moebius(n)/10.^n) \\ Charles R Greathouse IV, Feb 13 2011

Corrected by Charles R Greathouse IV, May 27 2015

A238273 Decimal expansion of the Sum_{n>=1} mu(n)/n^n where mu is the Moebius function.

Original entry on oeis.org

7, 1, 2, 6, 6, 3, 1, 8, 2, 2, 6, 4, 2, 8, 3, 4, 5, 7, 1, 2, 1, 6, 0, 4, 1, 6, 8, 3, 2, 1, 8, 6, 1, 2, 5, 0, 6, 0, 5, 3, 0, 7, 0, 2, 1, 7, 4, 7, 1, 6, 1, 3, 6, 9, 0, 7, 0, 3, 5, 0, 5, 0, 6, 3, 2, 9, 5, 5, 5, 6, 6, 1, 2, 1, 0, 4, 4, 7, 0, 6, 1, 4, 3, 6, 4, 1, 1, 4, 6, 2, 7, 0, 5, 9, 4, 3, 6, 2, 7, 6, 2, 1, 6, 0, 8

Offset: 0

Robert G. Wilson v, Feb 21 2014

Suggested by a posting to SeqFan list by Alonso Del Arte at 15:35 on Aug 09 2012.

			0.71266318226428345712160416832186125060530702174716136907035050632955566121...

Cf. A008683, A238270, A238271 & A181673.

RealDigits[ Sum[ N[ MoebiusMu[n] 1 n^-n, 128], {n, 100}], 10, 111][[1]]

A380387 Decimal expansion of Sum_{n>=1} lambda(n)/2^n where lambda is the Liouville function (A008836).

Original entry on oeis.org

1, 6, 2, 3, 2, 4, 6, 3, 7, 6, 2, 3, 9, 2, 9, 7, 8, 5, 9, 5, 9, 7, 9, 7, 3, 3, 5, 8, 3, 6, 2, 2, 4, 0, 9, 1, 7, 0, 0, 0, 4, 8, 3, 7, 9, 2, 2, 8, 8, 1, 5, 9, 4, 9, 1, 9, 3, 4, 4, 2, 9, 3, 8, 9, 7, 0, 7, 8, 5, 3, 2, 4, 7, 6, 9, 0, 1, 9, 5, 2, 4, 7, 9, 7, 2, 0

Offset: 0

Friedjof Tellkamp, Jan 23 2025

Close in value to log(zeta(Pi)) = 0.162324388... (A300892).

			0.16232463762392978595979733583622409170004837922881594919344293897...

Cf. A008836, A238270, A300892, A343784.

RealDigits[Sum[N[LiouvilleLambda[n]/ 2^n, 128], {n, 300}], 10, 86][[1]]

suminf(n=1, (-1)^bigomega(n)/2^n) \\ Michel Marcus, Jan 23 2025

Equals 2*A343784 - 1. - Hugo Pfoertner, Jan 25 2025

cp's OEIS Frontend

A238271 Decimal expansion of sum(n>=1, mu(n)/3^n ).

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A344432 a(n) = Sum_{k=1..n} mu(k) * 2^(n - k).

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A181673 Decimal expansion of Sum_{n>=1} moebius(n)*(1/10)^n.

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A238273 Decimal expansion of the Sum_{n>=1} mu(n)/n^n where mu is the Moebius function.

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A380387 Decimal expansion of Sum_{n>=1} lambda(n)/2^n where lambda is the Liouville function (A008836).

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