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A336939 Irregular triangle read by rows where T(n,k) is the number of divisors d of n! with k prime factors (counting multiplicity), such that both d and n!/d have distinct prime multiplicities.

%I A336939 #8 Aug 10 2020 00:24:26
%S A336939 1,1,1,1,0,2,0,1,2,0,2,1,0,2,0,0,2,0,1,2,1,2,2,1,2,1,0,2,0,2,0,2,0,2,
%T A336939 0,0,2,0,4,2,2,2,2,4,0,2,0,0,2,0,4,0,4,4,4,4,0,4,0,2,0,1,3,2,6,4,5,7,
%U A336939 6,6,7,5,4,6,2,3,1,0,2,0,4,2,2,4,4,4,4,4,2,2,4,0,2,0
%N A336939 Irregular triangle read by rows where T(n,k) is the number of divisors d of n! with k prime factors (counting multiplicity), such that both d and n!/d have distinct prime multiplicities.
%C A336939 A number has distinct prime multiplicities iff its prime signature is strict.
%e A336939 Triangle begins:
%e A336939   1
%e A336939   1
%e A336939   1 1
%e A336939   0 2 0
%e A336939   1 2 0 2 1
%e A336939   0 2 0 0 2 0
%e A336939   1 2 1 2 2 1 2 1
%e A336939   0 2 0 2 0 2 0 2 0
%e A336939   0 2 0 4 2 2 2 2 4 0 2 0
%e A336939   0 2 0 4 0 4 4 4 4 0 4 0 2 0
%e A336939   1 3 2 6 4 5 7 6 6 7 5 4 6 2 3 1
%e A336939 Row n = 8 counts the following divisors (empty columns shown as dots):
%e A336939   .  5  .  20  40   80  360   720   640  .  5760  .
%e A336939      7     28  56  112  504  1008   896     8064
%e A336939            45                      1440
%e A336939            63                      2016
%t A336939 Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n!/#]&&PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]
%Y A336939 A022559 gives row lengths minus one.
%Y A336939 A336500 is the generalization to all positive integers.
%Y A336939 A336868 gives the first (also last) column.
%Y A336939 A336869 gives row sums.
%Y A336939 A336870 is the version for superprimorials.
%Y A336939 A000005 counts divisors.
%Y A336939 A130091 lists numbers with distinct prime multiplicities.
%Y A336939 A181796 counts divisors with distinct prime multiplicities.
%Y A336939 A327498 gives the maximum divisor of n with distinct prime multiplicities.
%Y A336939 A336414 counts divisors of factorials with distinct prime multiplicities.
%Y A336939 A336415 counts divisors of factorials with equal prime multiplicities.
%Y A336939 A336423 counts chains using A130091.
%Y A336939 Cf. A006939, A098859, A118914, A124010, A336419, A336424, A336571, A336865.
%Y A336939 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336414, A336415, A336416.
%K A336939 nonn,tabf
%O A336939 0,6
%A A336939 _Gus Wiseman_, Aug 08 2020