A336619 - OEIS

A336619

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

%I #10 Dec 19 2023 09:19:39

%S 1,1,1,1,3,4,20,24,192,280,2800,17280,61600,207360,1976832,28028000,

%T 448448000,696729600,3811808000,12541132800,250822656000,

%U 5069704640000,111533502080000,115880067072000,2781121609728000,21277380032004096,447206762741760000

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

%C 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 After the first three terms, is this sequence strictly increasing?

%H Amiram Eldar, <a href="/A336619/b336619.txt">Table of n, a(n) for n = 0..500</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) = n!/A336618(n) = n!/A327526(n!).

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

%e 1: ()

%e 3: (1)

%e 4: (2)

%e 20: (2,1)

%e 24: (3,1)

%e 192: (6,1)

%e 280: (3,1,1)

%e 2800: (4,2,1)

%e 17280: (7,3,1)

%e 61600: (5,2,1,1)

%e 207360: (9,4,1)

%e 1976832: (9,3,1,1)

%e 28028000: (5,3,2,1,1)

%e 448448000: (9,3,2,1,1)

%e 696729600: (14,5,2,1)

%e 3811808000: (8,3,2,1,1,1)

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

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

%Y A336415 counts these divisors.

%Y A336617 is the version for distinct prime exponents.

%Y A336618 is the quotient n!/a(n).

%Y A047966 counts uniform partitions.

%Y A071625 counts distinct prime exponents.

%Y A072774 gives Heinz numbers of uniform partitions, with nonprime terms A182853.

%Y A130091 lists numbers with distinct prime exponents.

%Y A181796 counts divisors with distinct prime exponents.

%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, A118914, A124010, A327498, A327499, A336616.

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Jul 30 2020