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.

A334370 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k) / prime(k)!).

Original entry on oeis.org

1, 0, 1, 1, 0, 11, 0, 22, 56, 36, 2640, 1, 8712, 79, 72436, 360465, 48608, 49008961, 794376, 4232764, 7753140, 942565890, 18198334, 14799637777, 10577976, 366619314900, 2785137222400, 1475339135400, 1065920156634060, 3765722000041, 5869315258699050
Offset: 0

Views

Author

Ilya Gutkovskiy, May 11 2020

Keywords

Comments

a(n) is the number of functions f:[n]-> [n] such that the number of elements that are mapped to i is either 0 or the i-th prime. a(5) = 11: (33333), (11222), (12122), (12212), (12221), (21122), (21212), (21221), (22112), (22121), (22211). - Alois P. Heinz, Jul 18 2023

Links

Crossrefs

Cf. A000040, A000720, A007837, A032310, A115278, A178682, A190476, A319113, A364344.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      (p-> `if`(p>n, 0, b(n-p, i-1)*binomial(n, p)))(ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 18 2023
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, PrimeQ[#] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 30}]
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k!))) \\ Seiichi Manyama, Feb 27 2022

A305618 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).

Original entry on oeis.org

0, 1, 1, -3, -9, 20, 190, -126, -6280, -10326, 293041, 1519320, -16985045, -194560444, 1013712777, 27317463952, -19210030599, -4305097718760, -17733269020226, 743855089334604, 7868686621862292, -132351392654695270, -2854492900112993039, 20150897206881256464
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 06 2018

Keywords

Comments

Logarithmic transform of A010051.

Examples

			E.g.f.: A(x) = x^2/2! + x^3/3! - 3*x^4/4! - 9*x^5/5! + 20*x^6/6! + ...
exp(A(x)) = 1 + x^2/2! + x^3/3! + x^5/5! + x^7/7! + ... + x^A000040(k)/A039716(k) + ...
exp(exp(A(x))-1) = 1 + x^2/2! + x^3/3! + 3*x^4/4! + 11*x^5/5! + ... + A190476(k)*x^k/k! + ...

Links

Crossrefs

Cf. A000040, A002120, A007447, A010051, A039716, A190476, A218002, A303073, A305619.

Programs

  • Maple
    a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*t(n-j)*
       j*binomial(n, j), j=1..n-1)/n))(i-> `if`(isprime(i), 1, 0))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 04 2018
  • Mathematica
    nmax = 24; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[PrimeQ[n]] - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 24}]

A364450 Number of partitions of [n] without prime sized blocks.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 17, 43, 135, 536, 2262, 9109, 40119, 198069, 1057149, 5656915, 31937212, 191032078, 1218669125, 7948119483, 54117500635, 381631039690, 2828205076600, 21507011811289, 169880627954541, 1377653319819302, 11620433411120653, 100417638302823210
Offset: 0

Views

Author

Alois P. Heinz, Jul 25 2023

Keywords

Examples

			a(4) = 2: 1|2|3|4, 1234.

Links

Crossrefs

Cf. A000040, A097514, A190476, A329944.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      isprime(j), 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..30);

Formula

E.g.f.: exp(exp(x)-1-Sum_{p in primes} x^p/p!).

A347948 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)!).

Original entry on oeis.org

1, 0, 1, 1, 6, 21, 110, 673, 4312, 34260, 280212, 2648581, 26580840, 292468333, 3453838544, 43672689816, 590203920256, 8458029201433, 128528293405392, 2059731511022935, 34762308371221224, 615898019175914166, 11432263502365894916, 221855824404176472115
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2021

Keywords

Crossrefs

Cf. A000670, A010051, A023360, A190476, A329261, A347949.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Boole[PrimeQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A010051(k) * a(n-k).

A353163 Expansion of e.g.f. exp(Sum_{p prime} x^p / (p-1)!).

Original entry on oeis.org

1, 0, 2, 3, 12, 65, 210, 1477, 7560, 45864, 338310, 2176031, 17657640, 139280869, 1150004856, 10572694860, 94834041120, 931995595457, 9384294360168, 96974005210273, 1066116104926500, 11838081891521760, 137785102884102366, 1652584041236345933
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2022

Keywords

Crossrefs

Cf. A000040, A000248, A190476, A353162.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*x^k/(k-1)!))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k*binomial(n-1, k-1)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{p<=n, p prime} p * binomial(n-1,p-1) * a(n-p).

A329261 Expansion of e.g.f. -log(1 - Sum_{k>=1} x^prime(k) / prime(k)!).

Original entry on oeis.org

0, 0, 1, 1, 3, 11, 40, 232, 1246, 8912, 65766, 561001, 5198424, 52513111, 577791292, 6806860347, 86303601008, 1163845620633, 16701819148776, 253608108810052, 4065574363467636, 68608467057149112, 1215544196988580438, 22564088376584800717
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 09 2019

Keywords

Crossrefs

Cf. A000629, A010051, A190476, A305618, A305619.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[-Log[1 - Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Boole[PrimeQ[n]] + Sum[Binomial[n, k] Boole[PrimeQ[n - k]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

Formula

a(0) = 0; a(n) = A010051(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A010051(n-k) * k * a(k).

A351940 a(n) is the number of partitions of the set {1,2,...,n} into lists having a prime number of elements.

Original entry on oeis.org

1, 0, 2, 6, 12, 240, 480, 12600, 62160, 665280, 10009440, 94802400, 1497545280, 23662679040, 317854817280, 5236146115200, 102522189369600, 1772579589580800, 39459091697625600, 809304973699622400, 17463916757211724800, 388537548150495744000
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Examples

			a(2) =   2 : (12) (2! * 1 ways).
a(3) =   6 : (123) (3! * 1 ways).
a(4) =  12 : (12)(34) (2! * 2! * 3 ways).
a(5) = 240 : (12345) (5! * 1 ways), (123)(45) (3! * 2! * 10 ways).
a(6) = 480 : (123)(456) (3! * 3! * 10 ways), (12)(34)(56) (2! * 2! * 2! * 15 ways).

Crossrefs

Cf. A000262, A010051, A190476.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*x^k))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k!*binomial(n-1, k-1)*a(n-k)));

Formula

E.g.f.: Product_{k > 0} exp(x^prime(k)).
E.g.f.: exp(Sum_{k > 0} x^prime(k)).
a(0) = 1; a(n) = Sum_{p<=n, p prime} p! * binomial(n-1,p-1) * a(n-p).

A365107 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} x^prime(n) / prime(n)!^2 ).

Original entry on oeis.org

1, 0, 1, 1, 18, 101, 1550, 22492, 424536, 10283064, 272319552, 8959493401, 328044534576, 13799304374077, 657306569855728, 34694458662034731, 2048559070407831424, 132868259271772801185, 9463476338179250300352, 736376651361995115417850, 62178423492630241909006224, 5689134205956573233701281462
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 21 2023

Keywords

Crossrefs

Cf. A023998, A190476.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[x^Prime[k]/Prime[k]!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, Prime[k]]^2 Prime[k] a[n - Prime[k]], {k, 1, PrimePi[n]}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{p <= n, p prime} binomial(n,p)^2 * p * a(n-p).
Showing 1-8 of 8 results.

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