A292524 -id:A292524 - cp's OEIS frontend

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

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

A344433 a(n) = Sum_{k=1..n} mu(k) * k^(n - k).

Original entry on oeis.org

1, 0, -2, -6, -17, -46, -132, -402, -1314, -4613, -17313, -68893, -288556, -1269637, -5907157, -29489299, -160431708, -955478664, -6145884133, -41584238971, -287650358748, -1984825313901, -13377544470631, -86142095523089, -512881404732949, -2634567148684612, -9205461936290915, 17544751152746927

Offset: 1

Seiichi Manyama, May 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..598

Cf. A026898, A292524, A344432.

a[n_] := Sum[MoebiusMu[k] * k^(n-k), {k,1,n}]; Array[a, 30] (* Amiram Eldar, May 19 2021 *)

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

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

G.f.: Sum_{k>=1} mu(k) * x^k / (1 - k*x).

A292779 Interpret the values of the Moebius function mu(k) for k = n to 1 as a balanced ternary number.

Original entry on oeis.org

1, -2, -11, -11, -92, 151, -578, -578, -578, 19105, -39944, -39944, -571385, 1022938, 5805907, 5805907, -37240814, -37240814, -424661303, -424661303, 3062123098, 13522476301, -17858583308, -17858583308, -17858583308, 829430026135, 829430026135, 829430026135

Offset: 1

Alonso del Arte, Sep 22 2017

Balanced ternary is much like regular ternary, but with the crucial difference of using the digit -1 instead of the digit 2. Then some powers of 3 are added, others are subtracted.
Since the least significant digit is always 1, a(n) is never a multiple of 3.
If mu(n) = 0, then a(n) is the same as a(n - 1).
Run lengths are given by A076259. - Andrey Zabolotskiy, Oct 13 2017

			mu(1) = 1, so a(1) = 1 * 3^0 = 1.
mu(2) = -1, so a(2) = -1 * 3^1 + 1 * 3^0 = -3 + 1 = -2.
mu(3) = -1, so a(3) = -1 * 3^2 + -1 * 3^1 + 1 * 3^0 = -9 - 3 + 1 = -11.
mu(4) = 0, so a(4) = 0 * 3^3 + -1 * 3^2 + -1 * 3^1 + 1 * 3^0 = -9 - 3 + 1 = -11.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Wikipedia, Balanced ternary

Cf. A008683, A127513, A292524.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+3^(n-1)*numtheory[mobius](n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Oct 13 2017

Table[3^Range[0, n - 1].MoebiusMu[Range[n]], {n, 50}]

a(n) = sum(k=1, n, moebius(k)*3^(k-1)); \\ Michel Marcus, Oct 01 2017

a(n) = Sum_{k = 1 .. n} mu(k) 3^(k - 1).

A343425 a(n) = Sum_{k=1..n} mu(k) * n^(n - k).

Original entry on oeis.org

1, 1, 5, 44, 474, 6259, 98398, 1801784, 37726398, 889909001, 23363492888, 675898131588, 21367308429609, 732952005073611, 27116443849927291, 1076343749563379984, 45629840631648951966, 2057705657634136459302, 98357762859847238180913

Offset: 1

Seiichi Manyama, May 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A292524, A344431, A344432, A344433.

a[n_] := Sum[MoebiusMu[k] * n^(n-k), {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)

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

cp's OEIS Frontend

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

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A344433 a(n) = Sum_{k=1..n} mu(k) * k^(n - k).

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A292779 Interpret the values of the Moebius function mu(k) for k = n to 1 as a balanced ternary number.

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A343425 a(n) = Sum_{k=1..n} mu(k) * n^(n - k).

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