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A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

%I A336498 #7 Aug 06 2020 09:26:36
%S A336498 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 A336498 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 A336498 1,1,4,9,15,21,26,29,30,30,29,26,21,15,9,4,1
%N A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.
%C A336498 Row n is row n! of A146291. Row lengths are A022559(n) + 1.
%e A336498 Triangle begins:
%e A336498   1
%e A336498   1
%e A336498   1  1
%e A336498   1  2  1
%e A336498   1  2  2  2  1
%e A336498   1  3  4  4  3  1
%e A336498   1  3  5  6  6  5  3  1
%e A336498   1  4  8 11 12 11  8  4  1
%e A336498   1  4  8 11 12 12 12 12 11  8  4  1
%e A336498   1  4  8 12 16 19 20 20 19 16 12  8  4  1
%e A336498 Row n = 6 counts the following divisors:
%e A336498   1  2   4   8  16   48  144  720
%e A336498      3   6  12  24   72  240
%e A336498      5   9  18  36   80  360
%e A336498         10  20  40  120
%e A336498         15  30  60  180
%e A336498             45  90
%e A336498 Row n = 7 counts the following divisors:
%e A336498   1  2   4    8   16   48   144   720  5040
%e A336498      3   6   12   24   72   240  1008
%e A336498      5   9   18   36   80   336  1680
%e A336498      7  10   20   40  112   360  2520
%e A336498         14   28   56  120   504
%e A336498         15   30   60  168   560
%e A336498         21   42   84  180   840
%e A336498         35   45   90  252  1260
%e A336498              63  126  280
%e A336498              70  140  420
%e A336498             105  210  630
%e A336498                  315
%t A336498 Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]
%Y A336498 A000720 is column k = 1.
%Y A336498 A008302 is the version for superprimorials.
%Y A336498 A022559 gives row lengths minus one.
%Y A336498 A027423 gives row sums.
%Y A336498 A146291 is the generalization to non-factorials.
%Y A336498 A336499 is the restriction to divisors in A130091.
%Y A336498 A000142 lists factorial numbers.
%Y A336498 A336415 counts uniform divisors of n!.
%Y A336498 Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
%Y A336498 Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.
%K A336498 nonn,tabf
%O A336498 0,6
%A A336498 _Gus Wiseman_, Aug 03 2020