A336619 a(n) = n!/d where d is the maximum divisor of n! with equal prime exponents.

1, 1, 1, 1, 3, 4, 20, 24, 192, 280, 2800, 17280, 61600, 207360, 1976832, 28028000, 448448000, 696729600, 3811808000, 12541132800, 250822656000, 5069704640000, 111533502080000, 115880067072000, 2781121609728000, 21277380032004096, 447206762741760000

Offset: 0

Author

Gus Wiseman, Jul 30 2020

Keywords

• nonn

Comments

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

After the first three terms, is this sequence strictly increasing?

Examples

			The sequence of terms together with their prime signatures begins:
           1: ()
           1: ()
           1: ()
           1: ()
           3: (1)
           4: (2)
          20: (2,1)
          24: (3,1)
         192: (6,1)
         280: (3,1,1)
        2800: (4,2,1)
       17280: (7,3,1)
       61600: (5,2,1,1)
      207360: (9,4,1)
     1976832: (9,3,1,1)
    28028000: (5,3,2,1,1)
   448448000: (9,3,2,1,1)
   696729600: (14,5,2,1)
  3811808000: (8,3,2,1,1,1)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..500
• Gus Wiseman, Sequences counting and encoding certain classes of multisets.

Crossrefs

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

A336415 counts these divisors.

A336617 is the version for distinct prime exponents.

A336618 is the quotient n!/a(n).

A047966 counts uniform partitions.

A071625 counts distinct prime exponents.

A072774 gives Heinz numbers of uniform partitions, with nonprime terms A182853.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime exponents.

A319269 counts uniform factorizations.

A327524 counts factorizations of uniform numbers into uniform numbers.

A327527 counts uniform divisors.

Cf. A000005, A001222, A001597, A007916, A098859, A118914, A124010, A327498, A327499, A336616.

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

Programs

Mathematica

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

Formula

a(n) = n!/A336618(n) = n!/A327526(n!).