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