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A336618 Maximum divisor of n! with equal prime multiplicities.

%I A336618 #21 Dec 19 2023 09:19:47
%S A336618 1,1,2,6,8,30,36,210,210,1296,1296,2310,7776,30030,44100,46656,46656,
%T A336618 510510,1679616,9699690,9699690,10077696,10077696,223092870,223092870,
%U A336618 729000000,901800900,13060694016,13060694016,13060694016,78364164096,200560490130
%N A336618 Maximum divisor of n! with equal prime multiplicities.
%C A336618 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 A336618 Amiram Eldar, <a href="/A336618/b336618.txt">Table of n, a(n) for n = 0..1000</a>
%H A336618 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 A336618 a(n) = A327526(n!).
%e A336618 The sequence of terms together with their prime signatures begins:
%e A336618        1: ()
%e A336618        1: ()
%e A336618        2: (1)
%e A336618        6: (1,1)
%e A336618        8: (3)
%e A336618       30: (1,1,1)
%e A336618       36: (2,2)
%e A336618      210: (1,1,1,1)
%e A336618      210: (1,1,1,1)
%e A336618     1296: (4,4)
%e A336618     1296: (4,4)
%e A336618     2310: (1,1,1,1,1)
%e A336618     7776: (5,5)
%e A336618    30030: (1,1,1,1,1,1)
%e A336618    44100: (2,2,2,2)
%t A336618 Table[Max@@Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&],{n,0,15}]
%Y A336618 A327526 is the non-factorial generalization, with quotient A327528.
%Y A336618 A336415 counts these divisors.
%Y A336618 A336616 is the version for distinct prime multiplicities.
%Y A336618 A336619 is the quotient n!/a(n).
%Y A336618 A047966 counts uniform partitions.
%Y A336618 A071625 counts distinct prime multiplicities.
%Y A336618 A072774 lists uniform numbers.
%Y A336618 A130091 lists numbers with distinct prime multiplicities.
%Y A336618 A181796 counts divisors with distinct prime multiplicities.
%Y A336618 A319269 counts uniform factorizations.
%Y A336618 A327524 counts factorizations of uniform numbers into uniform numbers.
%Y A336618 A327527 counts uniform divisors.
%Y A336618 Cf. A000005, A001222, A001597, A007916, A098859, A124010, A182853, A327498.
%Y A336618 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.
%K A336618 nonn
%O A336618 0,3
%A A336618 _Gus Wiseman_, Jul 30 2020