cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Michael Somos, Jun 04 2000

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

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.

Programs

  • Haskell
    a055615 n = a008683 n * n  -- Reinhard Zumkeller, Sep 04 2015
  • Magma
    [n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014
  • Maple
    with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014
  • Mathematica
    Table[n MoebiusMu[n], {n,80}] (* Harvey P. Dale, May 26 2011 *)
  • PARI
    {a(n) = if( n<1, 0, n * moebius(n))};
  • PARI
    {a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n])};
  • Python
    from sympy import mobius
def A055615(n): return n*mobius(n) # Chai Wah Wu, Apr 01 2023
  • SageMath
    [n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022

Formula

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

Views

Author

Ilya Gutkovskiy, May 07 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A008683, A055615, A334659, A334660.
Cf. A000290, A046970, A053822.

Programs

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

Formula

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

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

Views

Author

Ilya Gutkovskiy, May 07 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A008683, A055615, A334657, A334660.
Cf. A000578, A053825, A063453.

Programs

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

Formula

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

A336278 a(n) = Sum_{k=1..n} mu(k)*k^4.

Original entry on oeis.org

1, -15, -96, -96, -721, 575, -1826, -1826, -1826, 8174, -6467, -6467, -35028, 3388, 54013, 54013, -29508, -29508, -159829, -159829, 34652, 268908, -10933, -10933, -10933, 446043, 446043, 446043, -261238, -1071238, -1994759, -1994759, -808838, 527498, 2028123
Offset: 1

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336276, A336277, A336279.
Cf. A008683, A055615, A070891, A344429, A344430.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^4, {k, #}] &, 35] (* Michael De Vlieger, Jul 15 2020 *)
Accumulate[Table[MoebiusMu[x]x^4,{x,40}]] (* Harvey P. Dale, Jan 14 2021 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^4); \\ Michel Marcus, Jul 15 2020
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A336278(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2*(j2**2*(j2*(6*j2 - 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30*A336278(k1)
        j, k1 = j2, n//j2
    return c-(n*(n**2*(n*(6*n + 15) + 10) - 1)-j*(j**2*(j*(6*j - 15) + 10) - 1))//30 # Chai Wah Wu, Apr 04 2023

Formula

Partial sums of A334660.
From Seiichi Manyama, Apr 03 2023: (Start)
G.f. A(x) satisfies x = Sum_{k>=1} k^4 * (1 - x^k) * A(x^k).
Sum_{k=1..n} k^4 * a(floor(n/k)) = 1. (End)
Showing 1-4 of 4 results.

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