A336616 Maximum divisor of n! with distinct prime multiplicities.

1, 1, 2, 3, 24, 40, 720, 1008, 8064, 72576, 3628800, 5702400, 68428800, 80870400, 317011968, 118879488000, 1902071808000, 2487324672000, 44771844096000, 50039119872000, 1000782397440000, 21016430346240000, 5085976143790080000, 6156707963535360000

Offset: 0

Author

Gus Wiseman, Jul 29 2020

Keywords

• nonn

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Examples

			The sequence of terms together with their prime signatures begins:
             1: ()
             1: ()
             2: (1)
             3: (1)
            24: (3,1)
            40: (3,1)
           720: (4,2,1)
          1008: (4,2,1)
          8064: (7,2,1)
         72576: (7,4,1)
       3628800: (8,4,2,1)
       5702400: (8,4,2,1)
      68428800: (10,5,2,1)
      80870400: (10,5,2,1)
     317011968: (11,5,2,1)
  118879488000: (11,6,3,2,1)
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..589
• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

A327498 is the version not restricted to factorials, with quotient A327499.

A336414 counts these divisors.

A336617 is the quotient n!/a(n).

A336618 is the version for equal prime multiplicities.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A327526 gives the maximum divisor of n with equal prime multiplicities.

A336415 counts divisors of n! with equal prime multiplicities.

Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Programs

Mathematica

Table[Max@@Select[Divisors[n!],UnsameQ@@Last/@If[#==1,{},FactorInteger[#]]&],{n,0,15}]

PARI

a(n) = { if(n < 2, return(1)); my(pr = primes(primepi(n)), res = pr[#pr]); for(i = 1, #pr, pr[i] = [pr[i], val(n, pr[i])] ); forstep(i = #pr, 2, -1, if(pr[i][2] < pr[i-1][2], res*=pr[i-1][1]^pr[i-1][2] ) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Aug 25 2020

Formula

a(n) = A327498(n!).