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A336616
Maximum divisor of n! with distinct prime multiplicities.
6
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
0,3
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.
FORMULA
a(n) = A327498(n!).
EXAMPLE
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)
MATHEMATICA
Table[Max@@Select[Divisors[n!], UnsameQ@@Last/@If[#==1, {}, FactorInteger[#]]&], {n, 0, 15}]
PROG
(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
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.
Sequence in context: A032811 A092049 A257789 * A354278 A061778 A160667
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2020
STATUS
approved