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A336619 a(n) = n!/d where d is the maximum divisor of n! with equal prime exponents.

%I A336619 #10 Dec 19 2023 09:19:39
%S A336619 1,1,1,1,3,4,20,24,192,280,2800,17280,61600,207360,1976832,28028000,
%T A336619 448448000,696729600,3811808000,12541132800,250822656000,
%U A336619 5069704640000,111533502080000,115880067072000,2781121609728000,21277380032004096,447206762741760000
%N A336619 a(n) = n!/d where d is the maximum divisor of n! with equal prime exponents.
%C A336619 A number has equal prime exponents iff it is a power of a squarefree number. We call such numbers uniform, so a(n) is n! divided by the maximum uniform divisor of n!.
%C A336619 After the first three terms, is this sequence strictly increasing?
%H A336619 Amiram Eldar, <a href="/A336619/b336619.txt">Table of n, a(n) for n = 0..500</a>
%H A336619 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 A336619 a(n) = n!/A336618(n) = n!/A327526(n!).
%e A336619 The sequence of terms together with their prime signatures begins:
%e A336619            1: ()
%e A336619            1: ()
%e A336619            1: ()
%e A336619            1: ()
%e A336619            3: (1)
%e A336619            4: (2)
%e A336619           20: (2,1)
%e A336619           24: (3,1)
%e A336619          192: (6,1)
%e A336619          280: (3,1,1)
%e A336619         2800: (4,2,1)
%e A336619        17280: (7,3,1)
%e A336619        61600: (5,2,1,1)
%e A336619       207360: (9,4,1)
%e A336619      1976832: (9,3,1,1)
%e A336619     28028000: (5,3,2,1,1)
%e A336619    448448000: (9,3,2,1,1)
%e A336619    696729600: (14,5,2,1)
%e A336619   3811808000: (8,3,2,1,1,1)
%t A336619 Table[n!/Max@@Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&],{n,0,15}]
%Y A336619 A327528 is the non-factorial generalization, with quotient A327526.
%Y A336619 A336415 counts these divisors.
%Y A336619 A336617 is the version for distinct prime exponents.
%Y A336619 A336618 is the quotient n!/a(n).
%Y A336619 A047966 counts uniform partitions.
%Y A336619 A071625 counts distinct prime exponents.
%Y A336619 A072774 gives Heinz numbers of uniform partitions, with nonprime terms A182853.
%Y A336619 A130091 lists numbers with distinct prime exponents.
%Y A336619 A181796 counts divisors with distinct prime exponents.
%Y A336619 A319269 counts uniform factorizations.
%Y A336619 A327524 counts factorizations of uniform numbers into uniform numbers.
%Y A336619 A327527 counts uniform divisors.
%Y A336619 Cf. A000005, A001222, A001597, A007916, A098859, A118914, A124010, A327498, A327499, A336616.
%Y A336619 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.
%K A336619 nonn
%O A336619 0,5
%A A336619 _Gus Wiseman_, Jul 30 2020