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Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.
5

%I #6 Aug 06 2020 09:27:00

%S 1,1,1,1,1,2,0,1,2,1,2,1,1,3,1,3,2,0,1,3,2,5,3,3,2,1,1,4,2,7,4,4,3,2,

%T 0,1,4,2,7,4,5,7,7,6,3,2,0,1,4,2,8,8,9,10,11,11,7,8,5,2,0,1,4,3,11,8,

%U 11,16,16,15,15,15,13,9,6,3,1,1,5,3,14,10,13,21,21,20,19,21,18,13,9,5,2,0

%N Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

%C Row lengths are A022559(n) + 1.

%e Triangle begins:

%e 1

%e 1

%e 1 1

%e 1 2 0

%e 1 2 1 2 1

%e 1 3 1 3 2 0

%e 1 3 2 5 3 3 2 1

%e 1 4 2 7 4 4 3 2 0

%e 1 4 2 7 4 5 7 7 6 3 2 0

%e 1 4 2 8 8 9 10 11 11 7 8 5 2 0

%e 1 4 3 11 8 11 16 16 15 15 15 13 9 6 3 1

%e 1 5 3 14 10 13 21 21 20 19 21 18 13 9 5 2 0

%e 1 5 3 14 10 14 25 23 27 24 30 28 28 25 20 16 11 5 2 0

%e Row n = 7 counts the following divisors:

%e 1 2 4 8 16 48 144 720 {}

%e 3 9 12 24 72 360 1008

%e 5 18 40 80 504

%e 7 20 56 112

%e 28

%e 45

%e 63

%t Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

%Y A000720 is column k = 1.

%Y A022559 gives row lengths minus one.

%Y A056172 appears to be column k = 2.

%Y A336414 gives row sums.

%Y A336420 is the version for superprimorials.

%Y A336498 is the version counting all divisors.

%Y A336865 is the generalization to non-factorials.

%Y A336866 lists indices of rows with a final 1.

%Y A336867 lists indices of rows with a final 0.

%Y A336868 gives the final terms in each row.

%Y A000110 counts divisors of superprimorials with distinct prime exponents.

%Y A008302 counts divisors of superprimorials by number of prime factors.

%Y A130091 lists numbers with distinct prime exponents.

%Y A181796 counts divisors with distinct prime exponents.

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

%Y Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.

%Y Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

%K nonn,tabf

%O 0,6

%A _Gus Wiseman_, Aug 03 2020