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.
LINKS
David A. Corneth, Table of n, a(n) for n = 0..589
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
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2020
STATUS
approved