A336618 Maximum divisor of n! with equal prime multiplicities.

1, 1, 2, 6, 8, 30, 36, 210, 210, 1296, 1296, 2310, 7776, 30030, 44100, 46656, 46656, 510510, 1679616, 9699690, 9699690, 10077696, 10077696, 223092870, 223092870, 729000000, 901800900, 13060694016, 13060694016, 13060694016, 78364164096, 200560490130

Offset

0,3

Comments

A number has equal prime multiplicities iff it is a power of a squarefree number. We call such numbers uniform, so a(n) is the maximum uniform divisor of n!.

Links

Amiram Eldar, Table of n, a(n) for n = 0..1000

Gus Wiseman, Sequences counting and encoding certain classes of multisets.

Formula

a(n) = A327526(n!).

Example

The sequence of terms together with their prime signatures begins:
       1: ()
       1: ()
       2: (1)
       6: (1,1)
       8: (3)
      30: (1,1,1)
      36: (2,2)
     210: (1,1,1,1)
     210: (1,1,1,1)
    1296: (4,4)
    1296: (4,4)
    2310: (1,1,1,1,1)
    7776: (5,5)
   30030: (1,1,1,1,1,1)
   44100: (2,2,2,2)

Mathematica

Table[Max@@Select[Divisors[n!], SameQ@@Last/@FactorInteger[#]&], {n, 0, 15}]

Crossrefs

A327526 is the non-factorial generalization, with quotient A327528.

A336415 counts these divisors.

A336616 is the version for distinct prime multiplicities.

A336619 is the quotient n!/a(n).

A047966 counts uniform partitions.

A071625 counts distinct prime multiplicities.

A072774 lists uniform numbers.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A319269 counts uniform factorizations.

A327524 counts factorizations of uniform numbers into uniform numbers.

A327527 counts uniform divisors.

Cf. A000005, A001222, A001597, A007916, A098859, A124010, A182853, A327498.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.

Sequence in context: A132269 A053287 A086323 * A075999 A096999 A019199

Adjacent sequences: A336615 A336616 A336617 * A336619 A336620 A336621

Keyword

nonn

Author

Gus Wiseman, Jul 30 2020

Status

approved