A218002 -id:A218002 - cp's OEIS frontend

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

1, 1, 2, 6, 18, 90, 420, 2220, 19020, 130860, 1096920, 9862920, 83843640, 1411202520, 16144792560, 203091829200, 2989264122000, 37012939750800, 597962683188000, 8681244913692000, 126467701221607200, 5006833609034743200, 95602098255580238400

Offset: 0

Alois P. Heinz, May 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..451

Cf. A000045, A193374, A205801, A218002, A272603, A273994, A317128.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(issqr(5*j^2+4) or issqr(5*j^2-4),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[5*j^2+4] || IntegerQ @ Sqrt[5*j^2-4], a[n-j]*(j-1)!*Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)

E.g.f.: exp(Sum_{n>=2} x^F(n)/F(n)) with F = A000045.

A317131 Number of permutations of [n] whose lengths of increasing runs are prime numbers.

Original entry on oeis.org

1, 0, 1, 1, 5, 19, 80, 520, 2898, 22486, 171460, 1509534, 14446457, 147241144, 1650934446, 19494460567, 248182635904, 3340565727176, 47659710452780, 718389090777485, 11381176852445592, 189580213656445309, 3305258537062221020, 60273557241570401742

Offset: 0

Alois P. Heinz, Jul 21 2018

nonn

			a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 19: 12345, 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000040, A097597, A218002, A317111, A317128, A317129, A317130, A317132, A317447.

g:= n-> `if`(n=0 or isprime(n), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);

g[n_] := If[n == 0 || PrimeQ[n], 1, 0];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

from functools import lru_cache
from sympy import isprime
def g(n): return int(n == 0 or isprime(n))
@lru_cache(maxsize=None)
def b(u, o, t):
  if u + o == 0: return g(t)
  return (sum(b(u-j,  o+j-1,  1) for j in range(1, u+1)) if g(t) else 0) +\
          sum(b(u+j-1, o-j, t+1) for j in range(1, o+1))
def a(n): return b(n, 0, 0)
print([a(n) for n in range(28)]) # Michael S. Branicky, Mar 29 2021 after Alois P. Heinz

A273998 Number of endofunctions on [n] whose cycle lengths are primes.

Original entry on oeis.org

1, 0, 1, 8, 75, 904, 13255, 229536, 4587961, 103971680, 2634212961, 73787255200, 2264440519891, 75563445303072, 2724356214102055, 105546202276277504, 4373078169296869425, 192970687573630633216, 9035613818754820178689, 447469496697658409400960

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000040, A060435, A116956, A218002, A273994, A273996, A273997, A305824.

b:= proc(n) option remember; local r, p;
      if n=0 then 1 else r, p:=0, 2;
      while p<=n do r:= r+(p-1)!*b(n-p)*
         binomial(n-1, p-1); p:= nextprime(p)
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, p}, If[n == 0, 1, {r, p} = {0, 2}; While[p <= n, r = r + (p - 1)!*b[n - p]*Binomial[n-1, p-1]; p = NextPrime[p]]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A329944 Number of permutations of [n] whose cycle lengths avoid primes.

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 211, 1051, 10081, 107857, 1227241, 8969401, 108817831, 1173362191, 19426473067, 320062090531, 5692838161921, 70426164947041, 1346222143950481, 21952313047471537, 493701484264143751, 10971915198235355071, 266542798822750395331

Offset: 0

Alois P. Heinz, Nov 24 2019

nonn

			a(4) = 7: (1)(2)(3)(4), (1234), (1243), (1324), (1342), (1423), (1432).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000040, A218002, A329945, A364450.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j), 0,
      a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[If[PrimeQ[j], 0,
     a[n-j] Binomial[n-1, j-1] (j-1)!], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz *)

a(n) mod 2 = 1.

a(n) mod 10 = period 5: repeat [1,1,1,1,7], g.f.: (7*x^4+x^3+x^2+x+1)/(1-x^5).

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

Ilya Gutkovskiy, Jun 06 2018

sign

Logarithmic transform of A010051.

			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! + ...

Alois P. Heinz, Table of n, a(n) for n = 1..472

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

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

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}]

A353162 Expansion of e.g.f. exp(Sum_{p prime} p * x^p).

Original entry on oeis.org

1, 0, 4, 18, 48, 1320, 4200, 115920, 994560, 11793600, 264055680, 2601244800, 67761429120, 1067726499840, 21513457405440, 485310649824000, 9925206939648000, 254012624170905600, 6174538264806912000, 160933619800835481600, 4458470291543671603200

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A082579, A218002, A351940, A353163, A353164.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*k*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, isprime(k)*k^2*a(n-k)/(n-k)!));

a(0) = 1; a(n) = (n-1)! * Sum_{p<=n, p prime} p^2 * a(n-p)/(n-p)!.

A305619 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)).

Original entry on oeis.org

0, 1, 2, -3, 4, -10, 636, -1078, -18416, -131976, 5035920, 5333592, 187347744, -4079616528, -14669908512, -140154110640, 28743506893056, -92449999037568, 2738959517576448, -52969092379214976, 34211286306178560, -16812071564735736576, 1407763084021569335808

Offset: 1

Ilya Gutkovskiy, Jun 06 2018

sign

			E.g.f.: A(x) = x^2/2! + 2*x^3/3! - 3*x^4/4! + 4*x^5/5! - 10*x^6/6! + ...
exp(A(x)) = 1 + x^2/2 + x^3/3 + x^5/5 + x^7/7 + ... + x^A000040(k)/A000040(k) + ...
exp(exp(A(x))-1) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + ... + A218002(k)*x^k/k! + ...

Cf. A000040, A002120, A007447, A218002, A305618.

a:=series(log(1+add(x^ithprime(k)/ithprime(k),k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; 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]] (n - 1)! - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]

A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 6, 64, 170, 2988, 14616, 180192, 1934712, 21673200, 300266736, 4220710272, 61785461712, 1003589762784, 17448621367680, 327598207658496, 6279739240655232, 134169095009652480, 2817563310900129024, 64570676279407718400, 1547773850801172960000, 38824156236466815920640

Offset: 0

Ilya Gutkovskiy, Sep 20 2021

nonn

Cf. A007840, A010051, A023360, A218002, A347948.

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] (k - 1)! Boole[PrimeQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

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

cp's OEIS Frontend

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

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Maple

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A317131 Number of permutations of [n] whose lengths of increasing runs are prime numbers.

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Maple

Mathematica

Python

A273998 Number of endofunctions on [n] whose cycle lengths are primes.

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Maple

Mathematica

A329944 Number of permutations of [n] whose cycle lengths avoid primes.

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Maple

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

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A353162 Expansion of e.g.f. exp(Sum_{p prime} p * x^p).

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PARI

PARI

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A305619 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)).

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Maple

Mathematica

A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

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Mathematica

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