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

Author

Gus Wiseman, Jul 30 2020

Keywords

• nonn

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!.

Examples

			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)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Sequences counting and encoding certain classes of multisets.

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.

Programs

Mathematica

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

Formula

a(n) = A327526(n!).