A334659 -id:A334659 - 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

A334657 Dirichlet g.f.: 1 / zeta(s-2).

Original entry on oeis.org

1, -4, -9, 0, -25, 36, -49, 0, 0, 100, -121, 0, -169, 196, 225, 0, -289, 0, -361, 0, 441, 484, -529, 0, 0, 676, 0, 0, -841, -900, -961, 0, 1089, 1156, 1225, 0, -1369, 1444, 1521, 0, -1681, -1764, -1849, 0, 0, 2116, -2209, 0, 0, 0, 2601, 0, -2809, 0, 3025, 0, 3249, 3364, -3481, 0, -3721

Offset: 1

Ilya Gutkovskiy, May 07 2020

Dirichlet inverse of A000290.
Moebius transform of A046970.
Inverse Moebius transform of A053822.

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

Cf. A008683, A055615, A334659, A334660.

Cf. A000290, A046970, A053822.

Mathematica
```
Table[MoebiusMu[n] n^2, {n, 61}]
```

G.f. A(x) satisfies: A(x) = x - 2^2 * A(x^2) - 3^2 * A(x^3) - 4^2 * A(x^4) - ...
a(1) = 1; a(n) = -n^2 * Sum_{d|n, d < n} a(d) / d^2.
a(n) = mu(n) * n^2.
Multiplicative with a(p^e) = -p^2 if e = 1 and 0 otherwise. - Amiram Eldar, Oct 25 2020

A336277 a(n) = Sum_{k=1..n} mu(k)*k^3.

Original entry on oeis.org

1, -7, -34, -34, -159, 57, -286, -286, -286, 714, -617, -617, -2814, -70, 3305, 3305, -1608, -1608, -8467, -8467, 794, 11442, -725, -725, -725, 16851, 16851, 16851, -7538, -34538, -64329, -64329, -28392, 10912, 53787, 53787, 3134, 58006, 117325, 117325, 48404

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, A336278, A336279.

Cf. A008683, A055615, A070891.

Array[Sum[MoebiusMu[k]*k^3, {k, #}] &, 41] (* Michael De Vlieger, Jul 15 2020 *)
Accumulate[Table[MoebiusMu[n] n^3,{n,50}]] (* Harvey P. Dale, Aug 15 2024 *)

a(n) = sum(k=1, n, moebius(k)*k^3); \\ Michel Marcus, Jul 15 2020

from functools import lru_cache
@lru_cache(maxsize=None)
def A336277(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A336277(k1)
        j, k1 = j2, n//j2
    return c-((n*(n+1))**2-((j-1)*j)**2>>2) # Chai Wah Wu, Apr 04 2023

Partial sums of A334659.
G.f. A(x) satisfies x = Sum_{k>=1} k^3 * (1 - x^k) * A(x^k). - Seiichi Manyama, Apr 01 2023
Sum_{k=1..n} k^3 * a(floor(n/k)) = 1. - Seiichi Manyama, Apr 03 2023

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

A359531 a(1) = 1, a(2) = -9; a(n) = -n^3 * Sum_{d|n, d < n} a(d) / d^3.

Original entry on oeis.org

1, -9, -27, 8, -125, 243, -343, 0, 0, 1125, -1331, -216, -2197, 3087, 3375, 0, -4913, 0, -6859, -1000, 9261, 11979, -12167, 0, 0, 19773, 0, -2744, -24389, -30375, -29791, 0, 35937, 44217, 42875, 0, -50653, 61731, 59319, 0, -68921, -83349, -79507, -10648, 0

Offset: 1

Seiichi Manyama, Apr 01 2023

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

Partial sums give A360658.
Cf. A092673, A359484, A359485.
Cf. A334659.

f[p_, e_] := If[e == 1, -p^3, 0]; f[2, e_] := Switch[e, 1, -9, 2, 8, , 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)= -9, a(4)= 8, a(2^e)= 0 if e>2. a(p)= -p^3, a(p^e)= 0 if e>1, p>2.

G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^3 * A(x^k).

A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

Original entry on oeis.org

1, 25, 217, 793, 3001, 5425, 16465, 25369, 52705, 75025, 159721, 172081, 369097, 411625, 651217, 811801, 1414945, 1317625, 2469241, 2379793, 3572905, 3993025, 6424177, 5505073, 9378001, 9227425, 12807289, 13056745, 20486761, 16280425, 28599361, 25977625, 34659457

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A084218, A350156.
Cf. A068970, A084220, A372964.
Cf. A001160, A008683.
Cf. A002117, A013664, A347328.

f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)
Dirichlet convolution of A334659 and A001160. - R. J. Mathar, Jul 14 2025

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

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A334657 Dirichlet g.f.: 1 / zeta(s-2).

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A336277 a(n) = Sum_{k=1..n} mu(k)*k^3.

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

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A359531 a(1) = 1, a(2) = -9; a(n) = -n^3 * Sum_{d|n, d < n} a(d) / d^3.

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A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

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