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 1-6 of 6 results.

A345030 a(n) = Sum_{k=1..n} n^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 70, 633, 7821, 117709, 2097684, 43047545, 1000010125, 25937439391, 743008621422, 23298085496173, 793714780786669, 29192926036832363, 1152921504875352376, 48661191876077295937, 2185911559749718388655, 104127350297928227579629
Offset: 1

Views

Author

Seiichi Manyama, Jun 06 2021

Keywords

Links

Crossrefs

Diagonal of A345032.
Cf. A023037, A332533, A344551, A345036.

Programs

  • Mathematica
    a[n_] := Sum[n^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 06 2021 *)
  • PARI
    a(n) = sum(k=1, n, n^(n\k-1));

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 - n*x^k).

A345037 a(n) = Sum_{k=1..n} (-k)^(floor(n/k) - 1).

Original entry on oeis.org

1, 0, 3, -1, 2, 3, 6, -12, 3, 20, 23, -49, -46, 41, 182, -100, -97, -6, -3, -613, 418, 1941, 1944, -5518, -4765, 1364, 10205, 2629, 2632, -1181, -1178, -71404, 7463, 105748, 127245, -233385, -233382, 159813, 868586, -335790, -335787, -853276, -853273, -2689757, 4163818
Offset: 1

Views

Author

Seiichi Manyama, Jun 06 2021

Keywords

Links

Crossrefs

Cf. A344551, A345036.

Programs

  • Mathematica
    a[n_] := Sum[(-k)^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 45] (* Amiram Eldar, Jun 06 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-k)^(n\k-1));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1+k*x^k))/(1-x))

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 + k*x^k).
|a(n)| ~ 3^((n - mod(n,3))/3 - 1). - Vaclav Kotesovec, Jun 12 2021

A345100 a(n) = Sum_{k=1..n} k^floor(n/k).

Original entry on oeis.org

1, 3, 6, 12, 17, 33, 40, 68, 95, 141, 152, 328, 341, 461, 738, 1130, 1147, 2159, 2178, 4068, 5841, 6997, 7020, 18198, 20723, 25001, 38798, 61546, 61575, 137445, 137476, 223252, 342593, 408435, 485376, 1213988, 1214025, 1476549, 2541498, 4202810, 4202851, 8777205
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2021

Keywords

Links

Crossrefs

Cf. A031971, A055225, A344551, A345098.

Programs

  • Mathematica
    a[n_] := Sum[k^Floor[n/k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 08 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^(n\k));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k*(1-x^k)/(1-k*x^k))/(1-x))

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} k*x^k * (1 - x^k)/(1 - k*x^k).
a(n) ~ 3^((n - mod(n,3))/3). - Vaclav Kotesovec, Jun 11 2021

A345094 a(n) = Sum_{k=1..n} floor(n/k)^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 68, 630, 7790, 117664, 2097224, 43046801, 1000000643, 25937425245, 743008378547, 23298085130341, 793714773371879, 29192926025508929, 1152921504608944840, 48661191875668966346, 2185911559738739586562, 104127350297911284587436
Offset: 1

Views

Author

Seiichi Manyama, Jun 07 2021

Keywords

Links

Crossrefs

Cf. A060946, A262843, A344551, A345030, A345098.

Programs

  • Mathematica
    a[n_] := Sum[Floor[n/k]^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)
  • PARI
    a(n) = sum(k=1, n, (n\k)^(n\k-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(j=1, N, (1-x^j)*sum(k=1, N, k^(k-1)*x^(j*k)))/(1-x))

Formula

G.f.: (1/(1 - x)) * Sum_{j>=1} Sum_{k>=1} k^(k-1) * x^(j*k) * (1 - x^j).

A345106 a(n) = Sum_{k=1..n} k^(n - floor(n/k)).

Original entry on oeis.org

1, 3, 14, 96, 971, 12015, 184286, 3283598, 67676125, 1572527901, 40843114146, 1170338862814, 36718016941445, 1251213685475261, 46033362584427670, 1818364700307111794, 76762441669319061911, 3448793841153099408185, 164309637864524321789042
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2021

Keywords

Links

Crossrefs

Cf. A076015, A344551, A345107.

Programs

  • Mathematica
    a[n_] := Sum[k^(n - Floor[n/k]), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^(n-n\k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k*(1-(k*x)^k)/((1-k^(k-1)*x^k)*(1-k*x))))

Formula

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

A344552 a(n) = Sum_{k=1..n} floor(k*(n-k)/n).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 6, 10, 12, 14, 17, 22, 25, 28, 34, 40, 46, 50, 55, 62, 69, 74, 80, 94, 100, 108, 115, 126, 133, 142, 152, 164, 176, 184, 197, 210, 221, 230, 242, 260, 272, 286, 297, 314, 327, 340, 354, 378, 394, 406, 423, 442, 459, 472, 488, 512, 532, 548, 563, 590, 607
Offset: 1

Views

Author

Wesley Ivan Hurt, May 22 2021

Keywords

Crossrefs

Cf. A006218, A344514, A344551.

Programs

  • Mathematica
    Table[Sum[Floor[k (n - k)/n], {k, n}], {n, 80}]
Showing 1-6 of 6 results.

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