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 1-4 of 4 results.

A157229 For the n in A157017, the number of representations of n! as the product of distinct numbers in the range n+1 to 2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 3, 17, 11, 10, 4, 4, 1, 3, 2, 8, 1, 7, 2, 1, 35, 5, 12, 2, 6, 108, 1, 18, 64, 4, 11, 103, 6, 16, 8, 36, 6, 10, 2, 237, 28, 1, 97, 30, 10, 165, 12, 42, 302, 32, 338, 120, 3, 1, 1983, 6, 6, 20, 3255, 9, 696, 882, 648, 6, 313, 13945, 360, 54, 288, 1419, 392, 94766
Offset: 1

Views

Author

T. D. Noe, Feb 25 2009; extended Feb 27 2009 with help from Ray Chandler

Keywords

Comments

See A157017 for examples and references.

A157612 Number of factorizations of n! into distinct factors.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 57, 253, 1060, 5285, 28762, 191263, 1052276, 8028450, 56576192, 424900240, 2584010916, 24952953943, 178322999025, 1886474434192, 15307571683248, 143131274598786, 1423606577935925, 17668243239613767, 137205093278725072, 1399239022852163764, 15774656316828338767
Offset: 0

Views

Author

Jaume Oliver Lafont, Mar 03 2009

Keywords

Comments

The number of factorizations of (n+1)! into k distinct factors can be arranged into the following triangle:
2! 1;
3! 1, 1;
4! 1, 3, 1;
5! 1, 7, 7, 1;
...

Examples

			3! = 6 = 2*3.
a(3) = 2 because there are 2 factorizations of 3!.
4! = 24 = 2*12 = 3*8 = 4*6 = 2*3*4.
a(4) = 5 because there are 5 factorizations of 4!.
5! = 120 (1)
5! = 2*60 = 3*40 = 4*30 = 5*24 = 6*20 = 8*15 = 10*12 (7)
5! = 2*3*20 = 2*4*15 = 2*5*12 = 2*6*10 = 3*4*10 = 3*5*8 = 4*5*6 (7)
5! = 2*3*4*5 (1)
a(5) = 16 because there are 16 factorizations of 5!.

Crossrefs

Cf. A076716, A157017, A157229, A318286. See A157836 for continuation of triangle.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(n!$2):
seq(a(n), n=0..12);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d-1]], {d, Divisors[n] ~Complement~ {1, n}}]];
a[n_] := b[n!, n!];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *)
  • PARI
    \\ See A318286 for count.
a(n)={if(n<=1, 1, count(factor(n!)[,2]))} \\ Andrew Howroyd, Feb 01 2020

Formula

a(n) = A045778(A000142(n)).

Extensions

a(8)-a(12) from Ray Chandler, Mar 07 2009
a(13)-a(17) from Alois P. Heinz, May 26 2013
a(18)-a(19) from Alois P. Heinz, Jan 10 2015
a(20)-a(26) from Andrew Howroyd, Feb 01 2020

A157625 Product of the composite numbers between n+1 and 2n, both inclusive.

Original entry on oeis.org

1, 4, 24, 48, 4320, 8640, 120960, 3628800, 7257600, 14515200, 6706022400, 13412044800, 8717829120000, 470762772480000, 941525544960000, 1883051089920000, 2112783322890240000, 147894832602316800000
Offset: 1

Views

Author

Jaume Oliver Lafont, Mar 03 2009

Keywords

Comments

This function is very useful in a problem due to Paul Erdős recorded in A157017. - M. F. Hasler, Feb 26 2014

Crossrefs

Cf. A073840, A157017, A144186 (product of primes between n+2 and 2n, both inclusive).

Programs

  • Mathematica
    nn=20;With[{comps=Complement[Range[2nn],Prime[Range[PrimePi[2nn]]]]}, Table[ Times@@ Select[comps,#>n&&#<=2n&],{n,nn}]] (* Harvey P. Dale, Feb 18 2013 *)
  • PARI
    a(n)=prod(i=n+1,2*n,if(isprime(i),1,i))

Formula

a(n) = n!*A000984(n)*A034386(n)/A034386(2n). - M. F. Hasler, Feb 26 2014

A193429 a(n) = minimum value of the largest element of a nonempty set of positive integers > n such that their product is equal to n!, or 0 if no such set exists.

Original entry on oeis.org

1, 0, 0, 6, 24, 12, 10, 20, 16, 28, 25, 22, 33, 30, 28, 28, 39, 35, 36, 44, 44, 42, 44, 50, 50, 50, 57, 57, 56, 58, 65, 64, 64, 72, 72, 70, 75, 80, 80, 78, 80, 88, 88, 86, 88, 95, 95, 94, 96, 102, 104, 102, 104, 111, 111, 110, 112, 120, 119, 118, 120, 122, 125
Offset: 0

Views

Author

William Rex Marshall, Jul 28 2011

Keywords

Comments

From Franklin T. Adams-Watters, Jul 28 2011: (Start)
For n > 4, there is always the factorization n! = (2*n) * (n!/(2*n)), so a(n) is only 0 for n = 1 or 2.
It appears that this sequence is O(n). (End)

Examples

			For n=5, n! = 120. Any factorization of 120 into 3 (or more) factors will have a factor <= 5, so we take the most central factorization into two factors, 120 = 10*12, the maximum of {10, 12} is 12, thus a(5) = 12.

Links

Crossrefs

Cf. A000142, A157017.
Showing 1-4 of 4 results.

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