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 41-45 of 45 results.

A359733 a(n) = (1/2) * Sum_{d|n} (2*d)^(n/d).

Original entry on oeis.org

1, 4, 7, 20, 21, 88, 71, 296, 373, 1084, 1035, 5084, 4109, 16496, 20787, 67728, 65553, 286516, 262163, 1070180, 1189937, 4194568, 4194327, 17760824, 16827241, 67109228, 72150655, 269503660, 268435485, 1104603808, 1073741855, 4303389216, 4476371181
Offset: 1

Views

Author

Seiichi Manyama, Jan 12 2023

Keywords

Crossrefs

Cf. A055225, A076717.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (2*#)^(n/#) &] / 2; Array[a, 33] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (2*d)^(n/d))/2;
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-2*k*x^k)))

Formula

G.f.: Sum_{k>0} k * x^k / (1 - 2 * k* x^k).

A363668 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n-1,d).

Original entry on oeis.org

1, 7, 19, 91, 151, 1135, 1765, 12355, 28846, 157917, 352837, 2280955, 5200469, 29986201, 80469589, 427061795, 1166803399, 6211188028, 17672632261, 89483074521, 271071666724, 1316291647997, 4116715364329, 19595444140771, 63205674328876, 292318539358879
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A055225, A362683, A363639, A363640.
Cf. A343549, A363669.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n - 1, #] &]; Array[a, 30] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n-1, d));

Formula

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

A309368 a(n) = Sum_{d|n} prime(n/d)^d.

Original entry on oeis.org

2, 7, 13, 32, 43, 129, 145, 405, 660, 1417, 2079, 5999, 8233, 18903, 37271, 74912, 131131, 300239, 524355, 1139985, 2180263, 4372491, 8388691, 17853809, 33715580, 68704969, 136183123, 274127445, 536871021, 1100025921, 2147483775, 4343912079, 8638792645, 17309012967, 34380645545
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 25 2019

Keywords

Crossrefs

Cf. A000040, A007445, A055225.

Programs

  • Mathematica
    Table[Sum[Prime[n/d]^d, {d, Divisors[n]}], {n, 1, 35}]
nmax = 35; CoefficientList[Series[Sum[Prime[k] x^k/(1 - Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 35; CoefficientList[Series[-Log[Product[(1 - Prime[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - prime(k)*x^k).
L.g.f.: -log(Product_{k>=1} (1 - prime(k)*x^k)^(1/k)).
a(n) ~ 2^n.

A326121 Expansion of Sum_{k>=1} k^2 * x^(2*k) / (1 - k * x^k).

Original entry on oeis.org

0, 1, 1, 5, 1, 18, 1, 33, 28, 58, 1, 246, 1, 178, 369, 577, 1, 1539, 1, 2774, 2531, 2170, 1, 16706, 3126, 8362, 20413, 35366, 1, 116444, 1, 135425, 178479, 131362, 94933, 1110999, 1, 524650, 1596521, 2530946, 1, 7280892, 1, 8403734, 16364457, 8389138, 1, 78568322, 823544, 43420683
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

Crossrefs

Cf. A055225, A078308, A087909.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^# &, # > 1 &], {n, 1, 50}]
  • PARI
    a(n)={sumdiv(n, d, if(d > 1, (n/d)^d))} \\ Andrew Howroyd, Sep 10 2019

Formula

a(n) = Sum_{d|n, d>1} (n/d)^d = Sum_{d|n, d
a(p) = 1, where p is prime.
a(n) = A055225(n) - n.

A382513 Expansion of Sum_{p prime} p * x^p / (1 - p * x^p).

Original entry on oeis.org

0, 2, 3, 4, 5, 17, 7, 16, 27, 57, 11, 145, 13, 177, 368, 256, 17, 1241, 19, 1649, 2530, 2169, 23, 10657, 3125, 8361, 19683, 18785, 29, 107442, 31, 65536, 178478, 131361, 94932, 793585, 37, 524649, 1596520, 1439201, 41, 6997770, 43, 4208945, 16302032
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 30 2025

Keywords

Crossrefs

Cf. A008472, A055225, A069359, A373458, A373459.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - Prime[k] x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

a(n) = Sum_{p|n, p prime} p^(n/p).
Previous Showing 41-45 of 45 results.

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