A336618 - OEIS

A336618

Maximum divisor of n! with equal prime multiplicities.

%I #21 Dec 19 2023 09:19:47

%S 1,1,2,6,8,30,36,210,210,1296,1296,2310,7776,30030,44100,46656,46656,

%T 510510,1679616,9699690,9699690,10077696,10077696,223092870,223092870,

%U 729000000,901800900,13060694016,13060694016,13060694016,78364164096,200560490130

%N Maximum divisor of n! with equal prime multiplicities.

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

%H Amiram Eldar, <a href="/A336618/b336618.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 a(n) = A327526(n!).

%e The sequence of terms together with their prime signatures begins:

%e 1: ()

%e 2: (1)

%e 6: (1,1)

%e 8: (3)

%e 30: (1,1,1)

%e 36: (2,2)

%e 210: (1,1,1,1)

%e 1296: (4,4)

%e 2310: (1,1,1,1,1)

%e 7776: (5,5)

%e 30030: (1,1,1,1,1,1)

%e 44100: (2,2,2,2)

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

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

%Y A336415 counts these divisors.

%Y A336616 is the version for distinct prime multiplicities.

%Y A336619 is the quotient n!/a(n).

%Y A047966 counts uniform partitions.

%Y A071625 counts distinct prime multiplicities.

%Y A072774 lists uniform numbers.

%Y A130091 lists numbers with distinct prime multiplicities.

%Y A181796 counts divisors with distinct prime multiplicities.

%Y A319269 counts uniform factorizations.

%Y A327524 counts factorizations of uniform numbers into uniform numbers.

%Y A327527 counts uniform divisors.

%Y Cf. A000005, A001222, A001597, A007916, A098859, A124010, A182853, A327498.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 30 2020