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.

Previous Showing 11-18 of 18 results.

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A059358.
Cf. A006218, A024916, A064602, A064603, A364970, A365439.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(n\k+3, 4));
  • Python
    from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023

A356038 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_2(k).

Original entry on oeis.org

1, 7, 28, 95, 286, 802, 2143, 5519, 13807, 33762, 81060, 191678, 447396, 1032647, 2360593, 5351231, 12041764, 26920297, 59829006, 132262550, 290990077, 637429514, 1390811841, 3023647046, 6551547161, 14151910442, 30481920523, 65480947739, 140318385088, 299995596747
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 24 2022

Keywords

Links

Crossrefs

Cf. A001157, A160399, A185003, A356039.
Cf. A006218, A024916, A064602, A064603.

Programs

  • Maple
    with(numtheory): seq(add(sigma[2](i)*binomial(n,i),i=1..n), n=1..60); # Ridouane Oudra, Oct 25 2022
  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[2, k], {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 2)); \\ Michel Marcus, Jul 24 2022

Formula

a(n) ~ zeta(3) * n^2 * 2^(n-2).
a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^2)*binomial(n,i*j). - Ridouane Oudra, Oct 25 2022

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

Original entry on oeis.org

1, 6, 34, 295, 3421, 50109, 873653, 17651130, 405071647, 10405074777, 295716745389, 9211817240589, 312086923832843, 11424093750214407, 449317984131076935, 18896062057857406028, 846136323944194170206, 40192544399241119212807
Offset: 1

Views

Author

Seiichi Manyama, Jul 20 2022

Keywords

Links

Crossrefs

Partial sums of A062796.
Cf. A006218, A024916, A064602, A064603, A064604, A248076, A319194.

Programs

  • PARI
    a(n) = sum(k=1, n, n\k*k^k);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, d^d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-x^k))/(1-x))
  • Python
    def A355887(n): return n*(1+n**(n-1))+sum(k**k*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} d^d.
G.f.: (1/(1-x)) * Sum_{k>0} (k * x)^k/(1 - x^k).

A356323 a(n) = n! * Sum_{k=1..n} sigma_3(k)/k.

Original entry on oeis.org

1, 11, 89, 794, 6994, 72204, 753108, 8973264, 111281616, 1524322080, 21601104480, 340803192960, 5483287025280, 96044874750720, 1748238132614400, 34093033838438400, 682396164763084800, 14706429413353574400, 323342442475011993600, 7585740483060676608000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A001158, A064603, A356010, A356297, A356298.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[3, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 3)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*(1+x^k)/(k*(1-x^k)^3))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k^2*log(1-x^k))/(1-x)))

Formula

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k * (1 + x^k)/(k * (1 - x^k)^3).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k^2 * log(1 - x^k).
a(n) ~ n! * Pi^4 * n^3 / 270. - Vaclav Kotesovec, Aug 07 2022

A366917 a(n) = Sum_{k=1..n} (-1)^k*k^3*floor(n/k).

Original entry on oeis.org

-1, 6, -22, 49, -77, 119, -225, 358, -399, 483, -849, 1139, -1059, 1349, -2179, 2500, -2414, 2885, -3975, 4971, -4661, 4663, -7505, 8819, -6932, 8454, -11986, 12438, -11952, 12744, -17048, 20399, -16897, 17501, -25843, 27904, -22750, 25270, -36274, 37184, -31738
Offset: 1

Views

Author

Chai Wah Wu, Oct 28 2023

Keywords

Links

Crossrefs

Cf. A024919, A366915, A064603.

Programs

  • Maple
    f:= proc(n) local k; add((-1)^k * k^3 * floor(n/k), k=1..n) end proc;
map(f, [$1..100]); # Robert Israel, Dec 29 2023
  • Mathematica
    a[n_]:=Sum[ (-1)^k*k^3*Floor[n/k],{k,n}]; Array[a,41] (* Stefano Spezia, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^k*k^3*(n\k)); \\ Michel Marcus, Oct 29 2023
  • Python
    from math import isqrt
def A366917(n): return (-(t:=isqrt(m:=n>>1))**3*(t+1)**2+sum((q:=m//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,t+1))<<2)+((s:=isqrt(n))**3*(s+1)**2 - sum((q:=n//k)*((k**3<<2)+q*(q*(q+2)+1)) for k in range(1,s+1))>>2)

Formula

a(n) = 16*A064603(floor(n/2)) - A064603(n).

A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

Original entry on oeis.org

1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760
Offset: 1

Views

Author

Seiichi Manyama, Jul 31 2022

Keywords

Links

Crossrefs

Column k=3 of A356250.
Cf. A000537, A024916, A318742, A350123.
Cf. A064603, A356125, A319086.

Programs

  • Mathematica
    a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    a(n) = sum(k=1, n, (k*(n\k))^3);
  • PARI
    a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))
  • Python
    from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
From Vaclav Kotesovec, Aug 02 2022: (Start)
a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).
a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)

A356039 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_3(k).

Original entry on oeis.org

1, 11, 58, 243, 866, 2804, 8485, 24387, 67333, 180086, 469338, 1196976, 2996956, 7385837, 17954243, 43125267, 102494548, 241309031, 563341508, 1305142418, 3002938045, 6866090880, 15609292379, 35299794600, 79443050541, 177989130174, 397124963671, 882642816697, 1954708794400
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 24 2022

Keywords

Comments

For m>0, Sum_{k=1..n} binomial(n,k) * sigma_m(k) ~ zeta(m+1) * n^m * 2^(n-m).

Links

Crossrefs

Cf. A001158, A160399, A185003, A356038.
Cf. A006218, A024916, A064602, A064603.

Programs

  • Maple
    with(numtheory): seq(add(sigma[3](i)*binomial(n,i), i=1..n), n=1..60); # Ridouane Oudra, Oct 31 2022
  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[3, k], {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 3)); \\ Michel Marcus, Jul 24 2022

Formula

a(n) ~ Pi^4 * n^3 * 2^(n-4) / 45.
a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^3)*binomial(n,i*j). - Ridouane Oudra, Oct 31 2022

A334064 Decimal expansion of Sum_{k>=1} 1/sigma_3(k).

Original entry on oeis.org

1, 1, 8, 3, 4, 7, 3, 5, 0, 6, 1, 7, 8, 8, 5, 1, 1, 0, 3, 8, 5, 7, 3, 4, 2, 3, 6, 8, 2, 9, 5, 2, 0, 9, 6, 9, 4, 5, 3, 6, 5, 7, 1, 0, 2, 1, 4, 0, 7, 1, 7, 5, 9, 7, 7, 1, 6, 8, 4, 7, 8, 8, 5, 3, 5, 4, 1, 7, 4, 1, 4, 4, 2, 1, 4, 3, 5, 3, 4, 7, 6, 7, 5, 4, 1, 0, 9, 2, 0, 5, 9, 7, 8, 3, 9, 4, 0, 9, 4, 9, 2, 2, 8, 5, 5
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 19 2020

Keywords

Examples

			1.183473506178851103857342368295209694536571021407175977168478853541741442...

Crossrefs

Cf. A001158, A064603, A109694.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; Do[Clear[f]; f[p_] := (1 + Sum[(p^3 - 1)/(p^(3 e + 3) - 1), {e, 1, emax}]); m = 1000; cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m+3}], x]]; Print[f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {emax, 100, 400, 100}]

Formula

Product_p Sum_{k>=0} 1/sigma_3(p^k).
Previous Showing 11-18 of 18 results.

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