A359531 -id:A359531 - cp's OEIS frontend

A359484 a(n) = n * mu(n) if n is odd, otherwise n * mu(n) - (n/2) * mu(n/2).

1, -3, -3, 2, -5, 9, -7, 0, 0, 15, -11, -6, -13, 21, 15, 0, -17, 0, -19, -10, 21, 33, -23, 0, 0, 39, 0, -14, -29, -45, -31, 0, 33, 51, 35, 0, -37, 57, 39, 0, -41, -63, -43, -22, 0, 69, -47, 0, 0, 0, 51, -26, -53, 0, 55, 0, 57, 87, -59, 30, -61, 93, 0, 0, 65, -99, -67, -34, 69, -105, -71, 0

Offset: 1

Seiichi Manyama, Mar 31 2023

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

Partial sums give A359478.
Cf. A008683, A055615, A358276.
Cf. A092673, A359485, A359531.

a[n_] := n * MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]*n/2]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

a(n) = n*moebius(n)-if(n%2, 0, n/2*moebius(n/2));

a(n) = A055615(n) if n is odd, otherwise A055615(n) - A055615(n/2).
a(n) is multiplicative with a(2)= -3, a(4)= 2, a(2^e)= 0 if e>2. a(p)= -p, a(p^e)= 0 if e>1, p>2.
a(1) = 1, a(2) = -3; a(n) = -n * Sum_{d|n, d < n} a(d) / d.
G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k * A(x^k).
a(n) = n*mu(n)-n*mu(n*2^(n mod 2)/2)*((n+1) mod 2)/2. - Wesley Ivan Hurt, Jun 09 2023

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).

A360658 a(1) = 1; a(n) = -Sum_{k=2..n} k^3 * a(floor(n/k)).

Original entry on oeis.org

1, -8, -35, -27, -152, 91, -252, -252, -252, 873, -458, -674, -2871, 216, 3591, 3591, -1322, -1322, -8181, -9181, 80, 12059, -108, -108, -108, 19665, 19665, 16921, -7468, -37843, -67634, -67634, -31697, 12520, 55395, 55395, 4742, 66473, 125792, 125792, 56871, -26478

Offset: 1

Seiichi Manyama, Apr 01 2023

sign

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

Partial sums of A359531.
Cf. A092149, A359478, A360390.
Cf. A336277.

f[p_, e_] := If[e == 1, -p^3, 0]; f[2, e_] := Switch[e, 1, -9, 2, 8, , 0]; s[1] = 1; s[n] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* Amiram Eldar, May 10 2023 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A360658(n):
    if n <= 1:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= ((j2*(j2-1))**2-(j*(j-1))**2>>2)*A360658(k1)
        j, k1 = j2, n//j2
    return c-((n*(n+1))**2-((j-1)*j)**2>>2) # Chai Wah Wu, Apr 01 2023

Sum_{k=1..n} k^3 * a(floor(n/k)) = 0 for n > 1.

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

cp's OEIS Frontend

A359484 a(n) = n * mu(n) if n is odd, otherwise n * mu(n) - (n/2) * mu(n/2).

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

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A360658 a(1) = 1; a(n) = -Sum_{k=2..n} k^3 * a(floor(n/k)).

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