A109102 -id:A109102 - cp's OEIS frontend

A325509 Number of factorizations of n! into factorial numbers > 1.

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 08 2019

nonn

			n = 10:
  (6*120*5040)
  (720*5040)
  (3628800)
n = 16:
  (2*2*2*2*1307674368000)
  (2*120*87178291200)
  (20922789888000)
n = 24:
  (2*2*6*25852016738884976640000)
  (24*25852016738884976640000)
  (620448401733239439360000)

Cf. A000142, A011371, A022559, A034876, A034878, A076934, A109100, A109101, A109102, A115627, A135291, A322583, A325272, A325273, A325276, A325508, A325510, A325511.

facs[n_,u_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d,u],Min@@#>=d&]],{d,Intersection[u,Rest[Divisors[n]]]}]];
Table[Length[facs[n!,Rest[Array[#!&,n]]]],{n,15}]

a(n) = 1 + A034876(n).

More terms from Alois P. Heinz, May 08 2019

A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

Original entry on oeis.org

10, 16, 24, 48, 96, 144, 192, 288, 384, 720, 768, 864, 1440, 1536, 1728, 3072, 4320, 5184, 6144, 8640, 10368, 12288, 24576, 25920, 31104, 40320, 49152, 51840, 62208, 80640, 86400, 98304

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7! = 6! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109100, A109101, A109102, A109103.

A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

Original entry on oeis.org

576, 1152, 2304, 2880, 3456, 4608, 5760, 6912, 9216, 11520, 18432, 20736, 23040, 36864, 41472, 46080, 73728, 92160

Offset: 1

Jud McCranie, Jun 19 2005

nonn

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109102, A109103.

A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

Original entry on oeis.org

4, 9, 288, 34560

Offset: 2

Jud McCranie, Jun 19 2005

nonn

			34560! = 2! * 3! * 4! * 5! * 34559!, using five different factorials, so a(5)=34560.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102.

A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 24, 32, 48, 72, 128, 192, 240, 384, 432, 480, 720, 864, 1152, 1440, 2592, 2880, 5040, 6144, 6912, 8192, 10080, 11520, 15360, 15552, 23040, 25920, 27648, 51840, 62208, 69120, 73728, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102, A109103.

cp's OEIS Frontend

A325509 Number of factorizations of n! into factorial numbers > 1.

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A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

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A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

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A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

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Keywords

Examples

Crossrefs

A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

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