A068340 -id:A068340 - cp's OEIS frontend

A336279 a(n) = Sum_{k=1..n} mu(k)*k^5.

1, -31, -274, -274, -3399, 4377, -12430, -12430, -12430, 87570, -73481, -73481, -444774, 93050, 852425, 852425, -567432, -567432, -3043531, -3043531, 1040570, 6194202, -242141, -242141, -242141, 11639235, 11639235, 11639235, -8871914, -33171914, -61801065

Offset: 1

Donald S. McDonald, Jul 15 2020

Conjecture: a(n) changes sign infinitely often.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002321, A068340, A336276, A336277, A336278.

Cf. A008683, A055615, A070891, A344429, A344430.

Array[Sum[MoebiusMu[k]*k^5, {k, #}] &, 32] (* Michael De Vlieger, Jul 15 2020 *)

a(n) = sum(k=1, n, moebius(k)*k^5); \\ Michel Marcus, Jul 15 2020

from functools import lru_cache
@lru_cache(maxsize=None)
def A336279(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2**2*(j2**2*(j2*(2*j2 - 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12*A336279(k1)
        j, k1 = j2, n//j2
    return c-(n**2*(n**2*(n*(2*n + 6) + 5) - 1)-j**2*(j**2*(j*(2*j - 6) + 5) - 1))//12 # Chai Wah Wu, Apr 04 2023

From Seiichi Manyama, Apr 03 2023: (Start)
G.f. A(x) satisfies x = Sum_{k>=1} k^5 * (1 - x^k) * A(x^k).
Sum_{k=1..n} k^5 * a(floor(n/k)) = 1. (End)

A361982 a(n) = 1 + Sum_{k=2..n} (-1)^k * k * a(floor(n/k)).

Original entry on oeis.org

1, 3, 0, 8, 3, -3, -10, 22, 22, 12, 1, -23, -36, -50, -35, 93, 76, 76, 57, 17, 38, 16, -7, -103, -103, -129, -129, -185, -214, -184, -215, 297, 330, 296, 331, 331, 294, 256, 295, 135, 94, 136, 93, 5, 5, -41, -88, -472, -472, -472, -421, -525, -578, -578, -523, -747, -690, -748

Offset: 1

Seiichi Manyama, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 1..8191

Partial sums of A332793.
Cf. A347030, A361983.
Cf. A068340.

from functools import lru_cache
@lru_cache(maxsize=None)
def A361982(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (((j2<<1)-1 if j2&1 else -(j2<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2)*A361982(k1)
        j, k1 = j2, n//j2
    return c+((-(n<<1)-1 if n&1 else (n<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2) # Chai Wah Wu, Apr 02 2023

Sum_{k=1..n} (-1)^k * k * a(floor(n/k)) = -1.

G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k * (1 - x^k) * A(x^k).

A344429 a(n) = Sum_{k=1..n} mu(k) * k^n.

Original entry on oeis.org

1, -3, -34, -96, -3399, 30239, -624046, -4482626, -32249230, 9768165230, -186975207617, -2150337557747, -327482869358214, 6894274639051756, 539094536846680025, 8044964790023844733, -707278869236116107432, -12275330572755863672628, -2190860499375418948848067

Offset: 1

Seiichi Manyama, May 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A002321, A008683, A031971, A068340, A321222, A332468, A336276, A336277, A336278, A336279, A344430.

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

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

from functools import lru_cache
from math import comb
from sympy import bernoulli
@lru_cache(maxsize=None)
def faulhaber(n,p):
    """ Faulhaber's formula for calculating Sum_{k=1..n} k^p
        requires sympy version 1.12+ where bernoulli(1) = 1/2
    """
    return sum(comb(p+1,k)*bernoulli(k)*n**(p-k+1) for k in range(p+1))//(p+1)
@lru_cache(maxsize=None)
def A344429(n,m=None):
    if n <= 1:
        return 1
    if m is None:
        m=n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (faulhaber(j-1,m)-faulhaber(j2-1,m))*A344429(k1,m)
        j, k1 = j2, n//j2
    return c+faulhaber(j-1,m)-faulhaber(n,m) # Chai Wah Wu, Nov 02 2023

A143158 Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j).

Original entry on oeis.org

1, 0, -1, -1, -2, -1, -1, -2, -1, 0, -2, -3, -2, -1, -1, -1, -2, -1, 0, 0, 1, -2, -3, -2, -1, -1, 0, -1, -2, -3, -2, -1, -1, 0, -1, 0, -2, -3, -2, -1, -1, 0, -1, 0, 0, -1, -2, -1, 0, 0, 1, 0, 1, 1, 1, -2, -3, -2, -1, -1, 0, -1, 0, 0, 0, -1, -2, -3, -2, -1, -1, 0, -1, 0, 0, 0, -1, 0, -3, -4, -3, -2, -2, -1, -2, -1, -1, -1, -2, -1, -1, -2, -3, -2, -1

Offset: 1

Gary W. Adamson, Jul 27 2008

Right border gives A008683.
Left border gives A002321.
Row sums give A068340.

			First few rows of the triangle =
   1;
   0, -1;
  -1, -2, -1;
  -1, -2, -1,  0;
  -2, -3, -2, -1, -1;
  -2, -3, -2, -1, -1, 0, -1;
  -2, -3, -2, -1, -1, 0, -1, 0;
  -1, -2, -1,  0,  0, 1,  0, 1, 1, 1;
  ...
For example, T(5,3) = (-2) = Sum(-1, 0, -1), since mu(n) = 1, -1, -1, 0, -1, ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A008683, A002321, A068340, A054527.

import Data.List (tails)
a143158 n k = a143158_tabl !! (n-1) !! (k-1)
a143158_row n = a143158_tabl !! (n-1)
a143158_tabl = map (map sum . init . tails) a054527_tabl
-- Reinhard Zumkeller, Sep 04 2015

Table[Sum[MoebiusMu@ j, {j, k, n}], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Dec 17 2015 *)

PARI
```
T(n,k) = sum(j=k,n,moebius(j))
```

Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j), where mu(n) = A008683.

T(n, k) = A000012(n) * (A008683(n) * 0^(n-k)) * A000012(n).

47th term = T(10,2) corrected by Reinhard Zumkeller, Sep 04 2015

A362021 a(n) = Sum_{k=1..n} (-1)^(n-k) * k * mu(k), where mu(k) is the Moebius function.

Original entry on oeis.org

1, -3, 0, 0, -5, 11, -18, 18, -18, 28, -39, 39, -52, 66, -51, 51, -68, 68, -87, 87, -66, 88, -111, 111, -111, 137, -137, 137, -166, 136, -167, 167, -134, 168, -133, 133, -170, 208, -169, 169, -210, 168, -211, 211, -211, 257, -304, 304, -304, 304, -253, 253, -306, 306, -251, 251, -194, 252, -311

Offset: 1

Seiichi Manyama, Apr 04 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A008683, A055615, A068340, A247418, A361982.

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

G.f.: (Sum_{k>=1} k * mu(k) * x^k) / (1 + x).
G.f. A(x) satisfies x = Sum_{k>=1} k * (1 + x^k) * A(x^k).
a(n) = -a(n-1) + A055615(n) for n > 1.

A068338 a(n) = n!^2 * Sum_{k=1..n} mu(k)/k^2, where mu(k) is the Moebius function.

Original entry on oeis.org

1, 3, 23, 368, 8624, 324864, 15399936, 985595904, 79833268224, 8115008716800, 968747865292800, 139499692602163200, 23346005516963020800, 4614592869368384716800, 1045883450064438558720000, 267746163216496271032320000, 76940878032870027275796480000

Offset: 1

Leroy Quet, Feb 27 2002

nonn

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

Cf. A008683, A068101, A068107, A068340.

A068338(n) = {n!^2 * sum(k=1,n,moebius(k)/k^2)} \\ John Tyler Rascoe, May 14 2025

a(16) onwards from John Tyler Rascoe, May 14 2025

A174651 Numbers n such that Sum_{k=1..n} k*Mobius(k) = n.

Original entry on oeis.org

1, 88, 125, 5246

Offset: 1

Carl Najafi, Dec 22 2012

nonn

No more terms < 10^7.

Cf. A008683, A002321, A055615, A068340.

muSums = Accumulate@Table[MoebiusMu[k] k, {k, 10^7}]; A174651 = {}; For[i = 1, i <= Length@muSums, i++, If[muSums[[i]] == i, AppendTo[A174651, i]]]; A174651
smQ[x_]:=x==Total[Table[n*MoebiusMu[n],{n,x}]]; Select[Range[6000],smQ] (* Harvey P. Dale, Oct 05 2019 *)

cp's OEIS Frontend

A336279 a(n) = Sum_{k=1..n} mu(k)*k^5.

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A361982 a(n) = 1 + Sum_{k=2..n} (-1)^k * k * a(floor(n/k)).

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A344429 a(n) = Sum_{k=1..n} mu(k) * k^n.

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A143158 Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j).

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A362021 a(n) = Sum_{k=1..n} (-1)^(n-k) * k * mu(k), where mu(k) is the Moebius function.

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A068338 a(n) = n!^2 * Sum_{k=1..n} mu(k)/k^2, where mu(k) is the Moebius function.

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A174651 Numbers n such that Sum_{k=1..n} k*Mobius(k) = n.

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