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 1-3 of 3 results.

A218002 E.g.f.: exp( Sum_{n>=1} x^prime(n) / prime(n) ).

Original entry on oeis.org

1, 0, 1, 2, 3, 44, 55, 1434, 3913, 39752, 392481, 5109290, 34683451, 914698212, 5777487703, 91494090674, 1504751645265, 31764834185744, 379862450767873, 12634073744624082, 132945783064464691, 2753044719709341980, 64135578414076991031, 1822831113987975441482
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2012

Keywords

Comments

Conjecture: a(n) = number of degree-n permutations of prime order.
The conjecture is false. Cf. A214003. This sequence gives the number of n-permutations whose cycle lengths are restricted to the prime numbers. - Geoffrey Critzer, Nov 08 2015

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + 55*x^6/6! + 1434*x^7/7! + ...
where
log(A(x)) = x^2/2 + x^3/3 + x^5/5 + x^7/7 + x^11/11 + x^13/13 + x^17/17 + x^19/19 + x^23/23 + x^29/29 + ... + x^prime(n)/prime(n) + ...
a(5) = 44 because there are 5!/5 = 24 permutations that are 5-cycles and there are 5!/(2*3) = 20 permutations that are the disjoint product of a 2-cycle and a 3-cycle. - _Geoffrey Critzer_, Nov 08 2015

Links

Crossrefs

Cf. A000040, A214003, A273001, A273998, A317131, A329944.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j),
      a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016
  • Mathematica
    f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Apply[And, PrimeQ[#]] &]]], {n, 0,23}] (* Geoffrey Critzer, Nov 08 2015 *)
  • PARI
    {a(n)=n!*polcoeff(exp(sum(k=1,n,x^prime(k)/prime(k))+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

A329945 Number of permutations of [n] whose cycle lengths avoid squares.

Original entry on oeis.org

1, 0, 1, 2, 3, 44, 175, 1434, 12313, 59912, 1057761, 9211850, 118785931, 1702959972, 21390805423, 339381890834, 4027183717425, 89818053205904, 1477419923299393, 28377482210884242, 608128083110593171, 11954214606663753500, 269933818505222203311
Offset: 0

Views

Author

Alois P. Heinz, Nov 24 2019

Keywords

Links

Crossrefs

Cf. A000290, A059841, A205801, A271751, A329944.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j), 0,
      a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(a(n), n=0..25);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ@Sqrt[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 *)

Formula

a(n) mod 2 = 1 - (n mod 2) = A059841(n).
a(n) mod 10 = period 10: repeat [1,0,1,2,3,4,5,4,3,2] = A271751(n-1) for n>0.

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!).
Showing 1-3 of 3 results.

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