A361983 -id:A361983 - cp's OEIS frontend

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

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

A361982 a(n) = 1 + Sum_{k=2..n} (-1)^k * k * a(floor(n/k)).

Original entry on oeis.org

1, 3, 0, 8, 3, -3, -10, 22, 22, 12, 1, -23, -36, -50, -35, 93, 76, 76, 57, 17, 38, 16, -7, -103, -103, -129, -129, -185, -214, -184, -215, 297, 330, 296, 331, 331, 294, 256, 295, 135, 94, 136, 93, 5, 5, -41, -88, -472, -472, -472, -421, -525, -578, -578, -523, -747, -690, -748

Offset: 1

Seiichi Manyama, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 1..8191

Partial sums of A332793.
Cf. A347030, A361983.
Cf. A068340.

from functools import lru_cache
@lru_cache(maxsize=None)
def A361982(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (((j2<<1)-1 if j2&1 else -(j2<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2)*A361982(k1)
        j, k1 = j2, n//j2
    return c+((-(n<<1)-1 if n&1 else (n<<1)+1)+(-(j<<1)+1 if j&1 else (j<<1)-1)>>2) # Chai Wah Wu, Apr 02 2023

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

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

A361981 a(1) = 1; a(n) = Sum_{k=2..n} (-1)^k * k^2 * a(floor(n/k)).

Original entry on oeis.org

1, 4, -5, 23, -2, -29, -78, 146, 146, 71, -50, -302, -471, -618, -393, 1399, 1110, 1110, 749, 49, 490, 127, -402, -2418, -2418, -2925, -2925, -4297, -5138, -4463, -5424, 8912, 10001, 9134, 10359, 10359, 8990, 7907, 9428, 3828, 2147, 3470, 1621, -1767, -1767, -3354, -5563

Offset: 1

Seiichi Manyama, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 1..8191

Partial sums of A361986.
Cf. A309288, A359479.
Cf. A360390, A361983.

f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := If[e == 1, 3, 7*2^(3*e-4)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* Amiram Eldar, May 09 2023 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A361981(n):
    if n <= 1:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += ((j2*(j2-1) if j2&1 else -j2*(j2-1))+(-j*(j-1) if j&1 else j*(j-1))>>1)*A361981(k1)
        j, k1 = j2, n//j2
    return c+((-n*(n+1) if n&1 else n*(n+1))+(-j*(j-1) if j&1 else j*(j-1))>>1) # Chai Wah Wu, Apr 02 2023

Sum_{k=1..n} (-1)^k * k^2 * a(floor(n/k)) = 0 for n > 1.

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

cp's OEIS Frontend

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

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A361982 a(n) = 1 + Sum_{k=2..n} (-1)^k * k * a(floor(n/k)).

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

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Mathematica

Formula

A361981 a(1) = 1; a(n) = Sum_{k=2..n} (-1)^k * k^2 * a(floor(n/k)).

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Links

Crossrefs

Programs

Mathematica

Python

Formula