A068340 -id:A068340 - cp's OEIS frontend

A076542 Numbers n such that A068340(n) = +-1.

1, 2, 86, 89, 90, 133, 5235, 5236

Offset: 1

Zak Seidov, Oct 19 2002

Presumably A068340(n)=0 only for n=10; so I looked for n's such that A068340(n)=+/-s; here s=1 (and n<=800000). For s=6, there's no solution at all! For s=8, n=34 (only 1 solution); for s=7, n={55, 56} (2 solutions); for s=9, n={5, 51, 52} (3 solutions) (for n up to 800000).

Any subsequent terms are > 10^12. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A068340.

A077030 Numbers k such that A068340(k)=+/-2.

Original entry on oeis.org

122, 127, 128, 142, 5233

Offset: 1

Zak Seidov, Oct 21 2002

Any subsequent terms are > 10^12. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A068340, A076542, A077031, A077032, A077033.

isok(k) = abs(sum(i=1, k, i*moebius(i))) == 2; \\ Michel Marcus, Jan 14 2023

A077031 Numbers k such that A068340(k)=+/-3.

Original entry on oeis.org

6, 130, 5027, 5028

Offset: 1

Zak Seidov, Oct 21 2002

Any subsequent terms are > 10^12. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A068340, A076542, A077030, A077032, A077033.

isok(k) = abs(sum(i=1, k, i*moebius(i))) == 3; \\ Michel Marcus, Jan 14 2023

A077032 Numbers k such that A068340(k)=+/-4.

Original entry on oeis.org

3, 4, 137, 5023, 5024, 5025, 5030, 5031, 5032

Offset: 1

Zak Seidov, Oct 21 2002

Any subsequent terms are > 10^12. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A068340, A076542, A077030, A077031, A077033.

isok(k) = abs(sum(i=1, k, i*moebius(i))) == 4; \\ Michel Marcus, Jan 14 2023

A077033 Numbers k such that A068340(k) = +/-5.

Original entry on oeis.org

15, 16, 5034, 5241

Offset: 1

Zak Seidov, Oct 21 2002

Any subsequent terms are > 10^12. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A068340, A076542, A077030, A077031, A077032.

isok(k) = abs(sum(i=1, k, i*moebius(i))) == 5; \\ Michel Marcus, Jan 14 2023

A076599 Numbers n such that A068340(n) = +-10.

Original entry on oeis.org

7, 8, 9, 14, 21, 37

Offset: 1

Zak Seidov, Oct 21 2002

A068340, A076542, A077030-77033.

Cf. A068340, A076542, A077030, A077031, A077032, A077033.

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

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

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

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

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, A336279.

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

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 *)

a(n) = sum(k=1, n, moebius(k)*k^4); \\ Michel Marcus, Jul 15 2020

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

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)

cp's OEIS Frontend

A076542 Numbers n such that A068340(n) = +-1.

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A077030 Numbers k such that A068340(k)=+/-2.

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A077031 Numbers k such that A068340(k)=+/-3.

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A077032 Numbers k such that A068340(k)=+/-4.

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A077033 Numbers k such that A068340(k) = +/-5.

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A076599 Numbers n such that A068340(n) = +-10.

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

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

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