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.

A343573 a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205
Offset: 1

Views

Author

Seiichi Manyama, Apr 20 2021

Keywords

Links

Crossrefs

Cf. A023887, A157019, A157020, A324158, A324159, A338661, A339481, A339482, A339712, A343567, A343574.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-x^k))^k))

Formula

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

A338662 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 5, 10, 29, 26, 123, 50, 305, 352, 668, 122, 3844, 170, 2593, 9704, 13825, 290, 41598, 362, 118259, 107986, 33047, 530, 929102, 394376, 130744, 1203580, 2737415, 842, 9910225, 962, 13315073, 14199222, 2404670, 33547310, 136502007, 1370, 10555795, 168405072, 548460064, 1682
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A081543, A338663, A343574.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-k*x^k)^k-1))

Formula

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

A338658 a(n) = Sum_{d|n} d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 4, 6, 14, 10, 36, 14, 56, 48, 80, 22, 228, 26, 140, 240, 316, 34, 552, 38, 820, 546, 308, 46, 2088, 680, 416, 1044, 2156, 58, 4380, 62, 3248, 1782, 680, 4690, 9672, 74, 836, 2808, 14560, 82, 15456, 86, 9108, 17040, 1196, 94, 37704, 12110, 21420, 5916, 16068, 106, 44496
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Crossrefs

Cf. A081543, A343574.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d*binomial(d+n/d-1, d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-x^k)^(k+1)))

Formula

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

A360824 Expansion of Sum_{k>0} (k * x)^k / (1 - k * x^k)^(k+1).

Original entry on oeis.org

1, 6, 30, 284, 3130, 47082, 823550, 16782664, 387422928, 10000094720, 285311670622, 8916102486528, 302875106592266, 11112006871683606, 437893890382576560, 18446744074918103056, 827240261886336764194, 39346408075331452862196
Offset: 1

Views

Author

Seiichi Manyama, Feb 22 2023

Keywords

Crossrefs

Cf. A339712, A343574, A360823.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-k*x^k)^(k+1)))
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-1, d));

Formula

a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-1,d).
If p is prime, a(p) = p + p^p.

A360831 Expansion of Sum_{k>0} (k * x)^k / (1 - (k * x)^k)^(k+1).

Original entry on oeis.org

1, 6, 30, 308, 3130, 49962, 823550, 17107464, 387617328, 10058609120, 285311670622, 8931600297696, 302875106592266, 11117432610599574, 437894531752211760, 18449277498826162192, 827240261886336764194, 39347911865350001626164
Offset: 1

Views

Author

Seiichi Manyama, Feb 22 2023

Keywords

Crossrefs

Cf. A338661, A343574, A360795, A360824.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-(k*x)^k)^(k+1)))
  • PARI
    a(n) = sumdiv(n, d, d^n*binomial(d+n/d-1, d));

Formula

a(n) = Sum_{d|n} d^n * binomial(d+n/d-1,d).
If p is prime, a(p) = p + p^p.
Showing 1-5 of 5 results.

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

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