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Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.
6

%I #14 Nov 10 2024 05:09:19

%S 1,1,3,5,24,38,132,195,570,1588,4193,6086,14561,19232,37142,106479,

%T 207291,266871,549726,674330,1465399,3086598,5939574,7182133,12324512,

%U 28968994,46819193,82873443,165205159,196666406,350397910,406894074,593725529,1229814478,1853300600,4024414209,6049714096,6968090487,9700557121,16810076542,26339337285

%N Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

%H Max Alekseyev, <a href="/A336425/b336425.txt">Table of n, a(n) for n = 0..85</a>

%e The a(4) = 24 divisors of divisors:

%e 1/1 2/1 3/1 4/1 8/1 12/1 24/1

%e 2/2 3/3 4/2 8/2 12/2 24/2

%e 4/4 8/4 12/3 24/3

%e 8/8 12/4 24/4

%e 12/12 24/8

%e 24/12

%e 24/24

%t strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];

%t Table[Total[Cases[Divisors[n!],d_?strsigQ:>Count[Divisors[d],e_?strsigQ]]],{n,0,20}]

%Y A336422 is the non-factorial generalization.

%Y A130091 lists numbers with distinct prime exponents.

%Y A181796 counts divisors with distinct prime exponents.

%Y A327526 gives the maximum divisor of n with equal prime exponents.

%Y A327498 gives the maximum divisor of n with distinct prime exponents.

%Y A336414 counts divisors of n! with distinct prime exponents.

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

%Y A336423 counts chains in A130091, with maximal version A336569.

%Y Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568, A336570, A336571, A336865, A336866, A336869.

%Y Factorial numbers: A000142, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327499, A336416, A336418, A336617.

%K nonn

%O 0,3

%A _Gus Wiseman_, Aug 06 2020

%E Terms a(21) onward from _Max Alekseyev_, Nov 07 2024