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.

A309126 a(n) = n + 2^3 * floor(n/2^3) + 3^3 * floor(n/3^3) + 4^3 * floor(n/4^3) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 78, 79, 80, 81, 82, 91, 92, 93, 94, 95, 96, 97, 98, 107, 108, 109, 110, 111, 112, 113, 114, 123, 124, 125, 126, 127, 128, 156, 157, 166, 167, 168, 169, 170, 171, 172, 173, 246, 247, 248, 249, 250, 251, 252
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 13 2019

Keywords

Comments

Partial sums of A113061.

Links

Crossrefs

Cf. A013937, A024916, A113061, A309125, A309127.

Programs

  • Mathematica
    Table[Sum[k^3 Floor[n/k^3], {k, 1, n}], {n, 1, 70}]
nmax = 70; CoefficientList[Series[1/(1 - x) Sum[k^3 x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, k^3*(n\k^3)); \\ Seiichi Manyama, Aug 30 2021

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} k^3 * x^(k^3)/(1 - x^(k^3)).
a(n) ~ zeta(4/3)*n^(4/3)/4 - n/2. - Vaclav Kotesovec, Aug 30 2021

A309127 a(n) = n + 2^4 * floor(n/2^4) + 3^4 * floor(n/3^4) + 4^4 * floor(n/4^4) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 13 2019

Keywords

Comments

Partial sums of A300909.

Links

Crossrefs

Cf. A013938, A024916, A300909, A309125, A309126.

Programs

  • Mathematica
    Table[Sum[k^4 Floor[n/k^4], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[k^4 x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=1, n, k^4*(n\k^4)); \\ Seiichi Manyama, Aug 30 2021

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} k^4 * x^(k^4)/(1 - x^(k^4)).
a(n) ~ zeta(5/4)*n^(5/4)/5 - n/2. - Vaclav Kotesovec, Aug 30 2021

A347397 a(n) = Sum_{k=1..n} k^k * floor(n/k^k).

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 48, 49, 50, 78, 83, 84, 85, 86, 91, 92, 93, 94, 99, 100, 101, 102, 107, 108, 109, 110, 115, 116, 117, 118, 123, 124, 125, 126, 131, 132, 160, 161, 166, 167, 168, 169, 174, 175, 176, 177, 182, 183, 184
Offset: 1

Views

Author

Seiichi Manyama, Aug 30 2021

Keywords

Comments

What is the limit_{n->infinity} a(n) / (n*log(n)/LambertW(log(n))) ?. - Vaclav Kotesovec, Aug 30 2021

Links

Crossrefs

Cf. A024916, A062071, A309125, A309126, A309127.

Programs

  • Mathematica
    Table[Sum[k^k*Floor[n/k^k], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 30 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^k*(n\k^k));

Formula

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

A384817 Numerator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 3, 17, 21, 25, 29, 17, 173, 191, 209, 463, 499, 535, 571, 2473, 2617, 2777, 2921, 3101, 3245, 3389, 3533, 3713, 96569, 100169, 34723, 36223, 37423, 38623, 39823, 20699, 21299, 21899, 22499, 69997, 71797, 73597, 75397, 77647, 79447, 81247, 83047, 85297
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2025

Keywords

Examples

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...

Crossrefs

Cf. A007406, A017667, A284648, A309125, A373439, A384818 (denominators).

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 44}] // Numerator
  • PARI
    a(n) = numerator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
a(n) / A384818(n) ~ Pi^4 * n / 90.

A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 2, 18, 18, 18, 36, 36, 36, 36, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 1200, 1200, 1200, 1200, 1200, 600, 600, 600, 600, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 600, 600, 600, 1200, 58800, 58800, 58800, 58800, 58800
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2025

Keywords

Examples

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...

Crossrefs

Cf. A007407, A017668, A284650, A309125, A373440, A384817 (numerators).

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 53}] // Denominator
  • PARI
    a(n) = denominator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).
a(n) is the denominator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
A384817(n) / a(n) ~ Pi^4 * n / 90.
Showing 1-5 of 5 results.

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