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.

A360390 a(1) = 1; a(n) = -Sum_{k=2..n} k^2 * a(floor(n/k)).

Original entry on oeis.org

1, -4, -13, -9, -34, 11, -38, -38, -38, 87, -34, -70, -239, 6, 231, 231, -58, -58, -419, -519, -78, 527, -2, -2, -2, 843, 843, 647, -194, -1319, -2280, -2280, -1191, 254, 1479, 1479, 110, 1915, 3436, 3436, 1755, -450, -2299, -2783, -2783, -138, -2347, -2347, -2347, -2347, 254, -422
Offset: 1

Views

Author

Seiichi Manyama, Apr 01 2023

Keywords

Links

Crossrefs

Partial sums of A359485.
Cf. A092149, A359478, A360658.
Cf. A336276.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, -p^2, 0]; f[2, e_] := Switch[e, 1, -5, 2, 4, , 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 A360390(n):
    if n <= 1:
        return 1
    c, j = 0, 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*A360390(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 01 2023

Formula

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

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Mar 31 2023

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

A361986 a(1) = 1, a(2) = 3; a(n) = n^2 * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d^2.

Original entry on oeis.org

1, 3, -9, 28, -25, -27, -49, 224, 0, -75, -121, -252, -169, -147, 225, 1792, -289, 0, -361, -700, 441, -363, -529, -2016, 0, -507, 0, -1372, -841, 675, -961, 14336, 1089, -867, 1225, 0, -1369, -1083, 1521, -5600, -1681, 1323, -1849, -3388, 0, -1587, -2209, -16128, 0, 0, 2601, -4732, -2809, 0, 3025, -10976
Offset: 1

Views

Author

Seiichi Manyama, Apr 02 2023

Keywords

Links

Crossrefs

Partial sums give A361981.
Cf. A361984, A361985.
Cf. A359485.

Programs

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

Formula

a(n) is multiplicative with a(2) = 3, a(2^e) = 7*2^(3*e-4) if e>1. a(p) = -p^2, a(p^e) = 0 if e>1, p>2.
G.f. A(x) satisfies -x * (1 - x) = Sum_{k>=1} (-1)^k * k^2 * A(x^k).

A359531 a(1) = 1, a(2) = -9; a(n) = -n^3 * Sum_{d|n, d < n} a(d) / d^3.

Original entry on oeis.org

1, -9, -27, 8, -125, 243, -343, 0, 0, 1125, -1331, -216, -2197, 3087, 3375, 0, -4913, 0, -6859, -1000, 9261, 11979, -12167, 0, 0, 19773, 0, -2744, -24389, -30375, -29791, 0, 35937, 44217, 42875, 0, -50653, 61731, 59319, 0, -68921, -83349, -79507, -10648, 0
Offset: 1

Views

Author

Seiichi Manyama, Apr 01 2023

Keywords

Links

Crossrefs

Partial sums give A360658.
Cf. A092673, A359484, A359485.
Cf. A334659.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, -p^3, 0]; f[2, e_] := Switch[e, 1, -9, 2, 8, , 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)= -9, a(4)= 8, a(2^e)= 0 if e>2. a(p)= -p^3, a(p^e)= 0 if e>1, p>2.
G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k^3 * A(x^k).
Showing 1-4 of 4 results.

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