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 1-8 of 8 results.

A080267 a(n) = Sum_{d divides n} d*2^(n-n/d).

Original entry on oeis.org

1, 5, 13, 41, 81, 257, 449, 1313, 2497, 6465, 11265, 33665, 53249, 143617, 269313, 672257, 1114113, 3159041, 4980737, 13568001, 23904257, 57675777, 96468993, 275980289, 424673281, 1090535425, 1963720705, 4823482369, 7784628225
Offset: 1

Views

Author

Vladeta Jovovic, Feb 11 2003

Keywords

Links

Crossrefs

Cf. A075900, A359112.

Programs

  • Maple
    oo := 40; s1 := add( k*2^(k-1)*x^k/(1-2^(k-1)*x^k),k=1..oo): s2 := series(s1,x,oo-1): s3 := seriestolist(%): A080267 := n->s3[n+1];
  • Mathematica
    a[n_] := Sum[d*2^(n-n/d), {d, Divisors[n]}]; Array[a, 29] (* Jean-François Alcover, Mar 20 2014 *)
  • PARI
    a(n) = sumdiv(n, d, d*2^(n-n/d)); \\ Michel Marcus, Mar 20 2014
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(2*x)^k)^2)) \\ Seiichi Manyama, Dec 20 2022

Formula

G.f.: Sum_{k>=1} k*2^(k-1)*x^k/(1 - 2^(k-1)*x^k). - N. J. A. Sloane, Jun 04 2003
G.f.: Sum_{k>=1} x^k/(1 - (2 * x)^k)^2. - Seiichi Manyama, Dec 20 2022

A359103 a(n) = Sum_{d|n} d * (n/d)^d.

Original entry on oeis.org

1, 4, 6, 16, 10, 54, 14, 112, 99, 230, 22, 996, 26, 1022, 1620, 3232, 34, 9828, 38, 18100, 16380, 22814, 46, 133272, 15675, 106886, 179388, 354116, 58, 1218150, 62, 1589824, 1952676, 2228870, 630980, 13767264, 74, 9962270, 20732868, 34787000, 82, 113676402, 86
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Links

Crossrefs

Cf. A055225, A087909, A167531, A359112, A359863.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^#*# &]; Array[a, 43] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*(n/d)^d);
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-k*x^k)^2))

Formula

a(n) = n * A087909(n).
G.f.: Sum_{k>=1} k * x^k/(1 - k * x^k)^2.
If p is prime, a(p) = 2 * p.
a(n) = [x^n] Sum_{k>0} k * (n * x / k)^k / (1 - x^k). - Seiichi Manyama, Jan 16 2023

A359203 a(n) = Sum_{d|n} (n/d) * 3^(n-d).

Original entry on oeis.org

1, 7, 28, 127, 406, 1756, 5104, 20575, 61237, 230122, 649540, 2579932, 6908734, 26044984, 74578888, 269985151, 731794258, 2799670555, 7360989292, 27392181562, 75948764752, 268482753172, 721764371008, 2742292424188, 7078172334031, 25701091008418, 71173405454680
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A000203, A080267, A115607, A359204.
Cf. A356539, A359112.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 3^(n-#)*n/# &]; Array[a, 27] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, n/d*3^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(3*x)^k)^2))

Formula

G.f.: Sum_{k>=1} x^k/(1 - (3 * x)^k)^2.

A359204 a(n) = Sum_{d|n} (n/d) * 4^(n-d).

Original entry on oeis.org

1, 9, 49, 289, 1281, 7041, 28673, 147969, 602113, 2951169, 11534337, 57876481, 218103809, 1056997377, 4113563649, 19394592769, 73014444033, 354385657857, 1305670057985, 6210524807169, 23571585826817, 108851659538433, 404620279021569, 1942025331015681
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A000203, A080267, A115607, A359203.
Cf. also A359112.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(n-#)*n/# &]; Array[a, 24] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, n/d*4^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(4*x)^k)^2))

Formula

G.f.: Sum_{k>=1} x^k/(1 - (4 * x)^k)^2.

A363664 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 56, 127, 1100, 1717, 19300, 64406, 383010, 352717, 23214660, 5200301, 191172406, 3465549077, 20859527460, 1166803111, 1010698826825, 17672631901, 102589250081802, 286539905316908, 75260204476154, 4116715363801, 548610025890719156
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A343548, A363662, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^(n+1).

A363667 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 37, 71, 751, 925, 13161, 45676, 262911, 184757, 18014557, 2704157, 133062875, 2838201061, 16907954129, 601080391, 830283170617, 9075135301, 87074953375981, 246003195539410, 53321730394923, 2104098963721, 479275771000215865, 1952680410445479976
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A363664, A363666.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^n.

A358660 a(n) = Sum_{d|n} d * (n/d)^(n-d).

Original entry on oeis.org

1, 4, 12, 76, 630, 7968, 117656, 2105416, 43048917, 1000781420, 25937424612, 743130116112, 23298085122494, 793742455829456, 29192926758107760, 1152930300766980112, 48661191875666868498, 2185915267189632382650, 104127350297911241532860
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Links

Crossrefs

Cf. A090879, A342629, A356539, A359112.

Programs

  • Mathematica
    a[n_] := Total[Map[#*(n/#)^(n - #) &, Divisors[n]]];
Table[a[n],{n,1,100}]
a[n_] := DivisorSum[n, (n/#)^(n-#)*# &]; Array[a, 19] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*(n/d)^(n-d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k/(1-k^(k-1)*x^k)^2))

Formula

G.f.: Sum_{k>=1} k^(k-1) * x^k/(1 - k^(k-1) * x^k)^2.
If p is prime, a(p) = p + p^(p-1).

A363649 Expansion of Sum_{k>0} x^(2*k)/(1 - (k*x)^k)^2.

Original entry on oeis.org

0, 1, 2, 4, 4, 14, 6, 56, 62, 266, 10, 3991, 12, 6158, 84996, 225296, 16, 2881607, 18, 96995583, 87740548, 2621462, 22, 30762215703, 122070312524, 50331674, 84457666628, 8631957089039, 28, 885639790229244, 30, 2814753793638432, 76826598191124
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A359112, A363646.
Cf. A363641.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-2*n/#) * (#-1) &]; Array[a, 33] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-2*n/d)*(d-1));

Formula

a(n) = Sum_{d|n} (n/d)^(n-2*n/d) * (d-1).
If p is prime, a(p) = p - 1.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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