A334657 -id:A334657 - cp's OEIS frontend

A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0

Offset: 1

Michael Somos, Jun 04 2000

Dirichlet inverse of n (A000027).
Absolute values give n if n is squarefree, otherwise 0.
a(n) is multiplicative because both mu(n) and n are. - Mitch Harris, Jun 09 2005
a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - David W. Wilson, Jun 12 2005
Negative of the Moebius number of the dihedral group of order 2n. - Eric M. Schmidt, Jul 28 2013

			G.f. = x - 2*x^2 - 3*x^3 - 5*x^5 + 6*x^6 - 7*x^7 + 10*x^10 - 11*x^11 - 13*x^13 + ...

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Mats Granvik, Primes approximated by eigenvalues.
Mats Granvik, Mobius function times n approximated by eigenvalues.

Moebius transform of A023900.
Cf. A000027 (Dirichlet inverse), A061669 (sum with it).
Cf. A062004.
Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).
Cf. A008683, A334657, A334659, A334660.

a055615 n = a008683 n * n  -- Reinhard Zumkeller, Sep 04 2015

[n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014

Table[n MoebiusMu[n], {n,80}] (* Harvey P. Dale, May 26 2011 *)

PARI
```
{a(n) = if( n<1, 0, n * moebius(n))};
```

{a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n])};

from sympy import mobius
def A055615(n): return n*mobius(n) # Chai Wah Wu, Apr 01 2023

[n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022

a(n) = n * A008683(n).
Dirichlet g.f.: 1/zeta(s-1).
Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller, Jul 17 2003
Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey, Sep 26 2004
Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2. - Amiram Eldar, Feb 02 2024

A336276 a(n) = Sum_{k=1..n} mu(k)*k^2.

Original entry on oeis.org

1, -3, -12, -12, -37, -1, -50, -50, -50, 50, -71, -71, -240, -44, 181, 181, -108, -108, -469, -469, -28, 456, -73, -73, -73, 603, 603, 603, -238, -1138, -2099, -2099, -1010, 146, 1371, 1371, 2, 1446, 2967, 2967, 1286, -478, -2327, -2327, -2327, -211, -2420

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, A336277, A336278, A336279.

Cf. A008683, A055615, A070891, A360390, A361983.

Array[Sum[MoebiusMu[k]*k^2, {k, #}] &, 47] (* Michael De Vlieger, Jul 15 2020 *)

a(n) = sum(k=1, n, moebius(k)*k^2); \\ Michel Marcus, Jul 15 2020

from functools import lru_cache
@lru_cache(maxsize=None)
def A336276(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2*(j2-1)*((j2<<1)-1)-j*(j-1)*((j<<1)-1))//6*A336276(k1)
        j, k1 = j2, n//j2
    return c-(n*(n+1)*((n<<1)+1)-j*(j-1)*((j<<1)-1))//6 # Chai Wah Wu, Apr 04 2023

Partial sums of A334657.
G.f. A(x) satisfies x = Sum_{k>=1} k^2 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^2 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023

A334659 Dirichlet g.f.: 1 / zeta(s-3).

Original entry on oeis.org

1, -8, -27, 0, -125, 216, -343, 0, 0, 1000, -1331, 0, -2197, 2744, 3375, 0, -4913, 0, -6859, 0, 9261, 10648, -12167, 0, 0, 17576, 0, 0, -24389, -27000, -29791, 0, 35937, 39304, 42875, 0, -50653, 54872, 59319, 0, -68921, -74088, -79507, 0, 0, 97336, -103823, 0, 0, 0, 132651, 0, -148877

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000578.
Moebius transform of A063453.
Inverse Moebius transform of A053825.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A055615, A334657, A334660.

Cf. A000578, A053825, A063453.

Mathematica
```
Table[MoebiusMu[n] n^3, {n, 53}]
```

G.f. A(x) satisfies: A(x) = x - 2^3 * A(x^2) - 3^3 * A(x^3) - 4^3 * A(x^4) - ...
a(1) = 1; a(n) = -n^3 * Sum_{d|n, d < n} a(d) / d^3.
a(n) = mu(n) * n^3.
Multiplicative with a(p^e) = -p^3 if e = 1 and 0 otherwise. - Amiram Eldar, Dec 05 2022

A334660 Dirichlet g.f.: 1 / zeta(s-4).

Original entry on oeis.org

1, -16, -81, 0, -625, 1296, -2401, 0, 0, 10000, -14641, 0, -28561, 38416, 50625, 0, -83521, 0, -130321, 0, 194481, 234256, -279841, 0, 0, 456976, 0, 0, -707281, -810000, -923521, 0, 1185921, 1336336, 1500625, 0, -1874161, 2085136, 2313441, 0, -2825761, -3111696, -3418801, 0, 0, 4477456

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000583.
Moebius transform of A189922.
Inverse Moebius transform of A053826.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A055615, A334657, A334659.

Cf. A000583, A053826, A189922.

Mathematica
```
Table[MoebiusMu[n] n^4, {n, 46}]
```

G.f. A(x) satisfies: A(x) = x - 2^4 * A(x^2) - 3^4 * A(x^3) - 4^4 * A(x^4) - ...
a(1) = 1; a(n) = -n^4 * Sum_{d|n, d < n} a(d) / d^4.
a(n) = mu(n) * n^4.
Multiplicative with a(p^e) = -p^4 if e = 1 and 0 otherwise. - Amiram Eldar, Dec 05 2022

A359485 a(1) = 1, a(2) = -5; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.

Original entry on oeis.org

1, -5, -9, 4, -25, 45, -49, 0, 0, 125, -121, -36, -169, 245, 225, 0, -289, 0, -361, -100, 441, 605, -529, 0, 0, 845, 0, -196, -841, -1125, -961, 0, 1089, 1445, 1225, 0, -1369, 1805, 1521, 0, -1681, -2205, -1849, -484, 0, 2645, -2209, 0, 0, 0, 2601, -676, -2809, 0, 3025, 0, 3249, 4205, -3481, 900, -3721, 4805, 0

Offset: 1

Seiichi Manyama, Apr 01 2023

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

Partial sums give A360390.
Cf. A092673, A359484, A359531.
Cf. A334657.

f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := Switch[e, 1, -5, 2, 4, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 10 2023 *)

a(n) is multiplicative with a(2)= -5, a(4)= 4, a(2^e)= 0 if e>2. a(p)= -p^2, a(p^e)= 0 if e>1, p>2.

G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^2 * A(x^k).

A361987 a(1) = 1; a(n) = n^2 * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d^2.

Original entry on oeis.org

1, 4, -9, 32, -25, -36, -49, 256, 0, -100, -121, -288, -169, -196, 225, 2048, -289, 0, -361, -800, 441, -484, -529, -2304, 0, -676, 0, -1568, -841, 900, -961, 16384, 1089, -1156, 1225, 0, -1369, -1444, 1521, -6400, -1681, 1764, -1849, -3872, 0, -2116, -2209, -18432, 0, 0, 2601, -5408, -2809, 0

Offset: 1

Seiichi Manyama, Apr 02 2023

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

Partial sums give A361983.
Cf. A067856, A332793.
Cf. A334657, A361986.

f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := 2^(3*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

a(n) is multiplicative with a(2^e) = 2^(3*e-1). a(p) = -p^2, a(p^e) = 0 if e>1, p>2.

G.f. A(x) satisfies -x = Sum_{k>=1} (-1)^k * k^2 * A(x^k).

A334879 a(1) = 1; a(n) = -(1/2) * Sum_{d|n, d > 1} d * (d + 1) * a(n/d).

Original entry on oeis.org

1, -3, -6, -1, -15, 15, -28, -3, -9, 35, -66, 6, -91, 63, 60, -9, -153, 27, -190, 15, 105, 143, -276, 21, -100, 195, -54, 28, -435, -75, -496, -27, 231, 323, 210, 18, -703, 399, 312, 55, -861, -105, -946, 66, 135, 575, -1128, 72, -441, 300, 510, 91, -1431, 189, 440

Offset: 1

Ilya Gutkovskiy, May 14 2020

sign

Dirichlet inverse of the positive triangular numbers A000217.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000217, A055615, A334657.

a[n_] := If[n == 1, n, -(1/2) Sum[If[d > 1, d (d + 1) a[n/d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 55}]

lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] =  -sumdiv(n, d, if (d>1, d*(d + 1)*va[n/d]))/2;); va;} \\ Michel Marcus, May 15 2020

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A000217(k) * A(x^k).

Dirichlet g.f.: 2 / (zeta(s-1) + zeta(s-2)).

A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

Original entry on oeis.org

1, -5, -11, 3, -29, 55, -55, 3, 16, 145, -131, -33, -181, 275, 319, 3, -305, -80, -379, -87, 605, 655, -551, -33, 96, 905, 16, -165, -869, -1595, -991, 3, 1441, 1525, 1595, 48, -1405, 1895, 1991, -87, -1721, -3025, -1891, -393, -464, 2755, -2255, -33, 288, -480, 3355, -543, -2861, -80, 3799

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Dirichlet inverse of A069097.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.

A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
A351654(n) = v351654[n]; \\ Antti Karttunen, Feb 16 2022

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).
a(n) = Sum_{d|n} A023900(n/d) * A334657(d).
a(n) = Sum_{d|n} A046970(n/d) * A055615(d).
a(n) = Sum_{d|n} A000005(n/d) * A328254(d).
Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - Amiram Eldar, May 25 2025

cp's OEIS Frontend

A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

A336276 a(n) = Sum_{k=1..n} mu(k)*k^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A334659 Dirichlet g.f.: 1 / zeta(s-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A334660 Dirichlet g.f.: 1 / zeta(s-4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A359485 a(1) = 1, a(2) = -5; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A361987 a(1) = 1; a(n) = n^2 * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A334879 a(1) = 1; a(n) = -(1/2) * Sum_{d|n, d > 1} d * (d + 1) * a(n/d).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula