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a(n) = n!/d where d = A336616(n) is the maximum divisor of n! with distinct prime multiplicities.
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%I #20 Sep 02 2020 23:06:04

%S 1,1,1,2,1,3,1,5,5,5,1,7,7,77,275,11,11,143,143,2431,2431,2431,221,

%T 4199,4199,4199,39083,39083,39083,898909,898909,26068361,26068361,

%U 215441,2141737,2141737,2141737,66393847,1009885357,7953594143,7953594143,294282983291

%N a(n) = n!/d where d = A336616(n) is the maximum divisor of n! with distinct prime multiplicities.

%C 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 Jinyuan Wang, <a href="/A336617/b336617.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) = A327499(n!).

%e The maximum divisor of 13! with distinct prime multiplicities is 80870400, so a(13) = 13!/80870400 = 77.

%t Table[n!/Max@@Select[Divisors[n!],UnsameQ@@Last/@If[#==1,{},FactorInteger[#]]&],{n,0,15}]

%Y A327499 is the non-factorial generalization, with quotient A327498.

%Y A336414 counts these divisors.

%Y A336616 is the maximum divisor d.

%Y A336619 is the version for equal prime multiplicities.

%Y A130091 lists numbers with distinct prime multiplicities.

%Y A181796 counts divisors with distinct prime multiplicities.

%Y A336415 counts divisors of n! with equal prime multiplicities.

%Y Cf. A000005, A098859, A124010, A336424, A336500, A336568.

%Y Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327500, A336416, A336618.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 29 2020

%E More terms from _Jinyuan Wang_, Jul 31 2020