cp's OEIS Frontend

A336616 Maximum divisor of n! with distinct prime multiplicities.

%I A336616 #21 Sep 02 2020 23:05:53
%S A336616 1,1,2,3,24,40,720,1008,8064,72576,3628800,5702400,68428800,80870400,
%T A336616 317011968,118879488000,1902071808000,2487324672000,44771844096000,
%U A336616 50039119872000,1000782397440000,21016430346240000,5085976143790080000,6156707963535360000
%N A336616 Maximum divisor of n! with distinct prime multiplicities.
%C A336616 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.
%H A336616 David A. Corneth, <a href="/A336616/b336616.txt">Table of n, a(n) for n = 0..589</a>
%H A336616 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%F A336616 a(n) = A327498(n!).
%e A336616 The sequence of terms together with their prime signatures begins:
%e A336616              1: ()
%e A336616              1: ()
%e A336616              2: (1)
%e A336616              3: (1)
%e A336616             24: (3,1)
%e A336616             40: (3,1)
%e A336616            720: (4,2,1)
%e A336616           1008: (4,2,1)
%e A336616           8064: (7,2,1)
%e A336616          72576: (7,4,1)
%e A336616        3628800: (8,4,2,1)
%e A336616        5702400: (8,4,2,1)
%e A336616       68428800: (10,5,2,1)
%e A336616       80870400: (10,5,2,1)
%e A336616      317011968: (11,5,2,1)
%e A336616   118879488000: (11,6,3,2,1)
%t A336616 Table[Max@@Select[Divisors[n!],UnsameQ@@Last/@If[#==1,{},FactorInteger[#]]&],{n,0,15}]
%o A336616 (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 }
%o A336616 val(n, p) = my(r=0); while(n, r+=n\=p); r \\ _David A. Corneth_, Aug 25 2020
%Y A336616 A327498 is the version not restricted to factorials, with quotient A327499.
%Y A336616 A336414 counts these divisors.
%Y A336616 A336617 is the quotient n!/a(n).
%Y A336616 A336618 is the version for equal prime multiplicities.
%Y A336616 A130091 lists numbers with distinct prime multiplicities.
%Y A336616 A181796 counts divisors with distinct prime multiplicities.
%Y A336616 A327526 gives the maximum divisor of n with equal prime multiplicities.
%Y A336616 A336415 counts divisors of n! with equal prime multiplicities.
%Y A336616 Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568.
%Y A336616 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.
%K A336616 nonn
%O A336616 0,3
%A A336616 _Gus Wiseman_, Jul 29 2020