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 1-5 of 5 results.

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

Views

Author

Donald S. McDonald, Jul 15 2020

Keywords

Comments

Conjecture: a(n) changes sign infinitely often.

Links

Crossrefs

Cf. A002321, A068340, A336277, A336278, A336279.
Cf. A008683, A055615, A070891, A360390, A361983.

Programs

  • Mathematica
    Array[Sum[MoebiusMu[k]*k^2, {k, #}] &, 47] (* Michael De Vlieger, Jul 15 2020 *)
  • PARI
    a(n) = sum(k=1, n, moebius(k)*k^2); \\ Michel Marcus, Jul 15 2020
  • Python
    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

Formula

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

A359478 a(1) = 1; a(n) = -Sum_{k=2..n} k * a(floor(n/k)).

Original entry on oeis.org

1, -2, -5, -3, -8, 1, -6, -6, -6, 9, -2, -8, -21, 0, 15, 15, -2, -2, -21, -31, -10, 23, 0, 0, 0, 39, 39, 25, -4, -49, -80, -80, -47, 4, 39, 39, 2, 59, 98, 98, 57, -6, -49, -71, -71, -2, -49, -49, -49, -49, 2, -24, -77, -77, -22, -22, 35, 122, 63, 93, 32, 125, 125, 125, 190, 91
Offset: 1

Views

Author

Seiichi Manyama, Mar 31 2023

Keywords

Links

Crossrefs

Partial sums of A359484.
Cf. A092149, A360390, A360658.
Cf. A359479.

Programs

  • Mathematica
    s[n_] := n * MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]*n/2]; Accumulate[Array[s, 100]] (* Amiram Eldar, May 09 2023 *)
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A359478(n):
    if n <= 1:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2*(j2-1)-j*(j-1)>>1)*A359478(k1)
        j, k1 = j2, n//j2
    return c-(n*(n+1)-(j-1)*j>>1) # Chai Wah Wu, Mar 31 2023

Formula

Sum_{k=1..n} k * a(floor(n/k)) = 0 for n > 1.
G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k * (1 - x^k) * A(x^k).

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

Views

Author

Seiichi Manyama, Apr 01 2023

Keywords

Links

Crossrefs

Partial sums give A360390.
Cf. A092673, A359484, A359531.
Cf. A334657.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Apr 01 2023

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

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).

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

Views

Author

Seiichi Manyama, Apr 02 2023

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

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).
Showing 1-5 of 5 results.

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