The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #7 Aug 06 2020 09:26:36
%S 1,1,1,1,1,2,1,1,2,2,2,1,1,3,4,4,3,1,1,3,5,6,6,5,3,1,1,4,8,11,12,11,8,
%T 4,1,1,4,8,11,12,12,12,12,11,8,4,1,1,4,8,12,16,19,20,20,19,16,12,8,4,
%U 1,1,4,9,15,21,26,29,30,30,29,26,21,15,9,4,1
%N Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.
%C Row n is row n! of A146291. Row lengths are A022559(n) + 1.
%e Triangle begins:
%e 1
%e 1
%e 1 1
%e 1 2 1
%e 1 2 2 2 1
%e 1 3 4 4 3 1
%e 1 3 5 6 6 5 3 1
%e 1 4 8 11 12 11 8 4 1
%e 1 4 8 11 12 12 12 12 11 8 4 1
%e 1 4 8 12 16 19 20 20 19 16 12 8 4 1
%e Row n = 6 counts the following divisors:
%e 1 2 4 8 16 48 144 720
%e 3 6 12 24 72 240
%e 5 9 18 36 80 360
%e 10 20 40 120
%e 15 30 60 180
%e 45 90
%e Row n = 7 counts the following divisors:
%e 1 2 4 8 16 48 144 720 5040
%e 3 6 12 24 72 240 1008
%e 5 9 18 36 80 336 1680
%e 7 10 20 40 112 360 2520
%e 14 28 56 120 504
%e 15 30 60 168 560
%e 21 42 84 180 840
%e 35 45 90 252 1260
%e 63 126 280
%e 70 140 420
%e 105 210 630
%e 315
%t Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]
%Y A000720 is column k = 1.
%Y A008302 is the version for superprimorials.
%Y A022559 gives row lengths minus one.
%Y A027423 gives row sums.
%Y A146291 is the generalization to non-factorials.
%Y A336499 is the restriction to divisors in A130091.
%Y A000142 lists factorial numbers.
%Y A336415 counts uniform divisors of n!.
%Y Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
%Y Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.
%K nonn,tabf
%O 0,6
%A _Gus Wiseman_, Aug 03 2020