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A336420 Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.

%I A336420 #17 Sep 02 2020 23:04:03
%S A336420 1,1,1,1,2,1,1,1,3,2,5,2,1,1,1,4,3,11,7,7,10,5,2,1,1,1,5,4,19,14,18,
%T A336420 37,25,23,15,23,10,5,2,1,1,1,6,5,29,23,33,87,70,78,74,129,84,81,49,39,
%U A336420 47,23,10,5,2,1,1,1,7,6,41,34,52,165,144,183,196,424,317,376,325,299,431,304,261,172,129,81,103,47,23,10,5,2,1,1
%N A336420 Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.
%C A336420 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
%C A336420 The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
%C A336420 T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.
%e A336420 Triangle begins:
%e A336420   1
%e A336420   1  1
%e A336420   1  2  1  1
%e A336420   1  3  2  5  2  1  1
%e A336420   1  4  3 11  7  7 10  5  2  1  1
%e A336420   1  5  4 19 14 18 37 25 23 15 23 10  5  2  1  1
%e A336420 The divisors counted in row n = 4 are:
%e A336420   1  2  4     8   16   48   144   432  2160  10800  75600
%e A336420      3  9    12   24   72   360   720  3024
%e A336420      5  25   18   40   80   400  1008
%e A336420      7       20   54  108   504  1200
%e A336420              27   56  112   540  2800
%e A336420              28  135  200   600
%e A336420              45  189  675   756
%e A336420              50            1350
%e A336420              63            1400
%e A336420              75            4725
%e A336420             175
%t A336420 chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t A336420 Table[Length[Select[Divisors[chern[n]],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,5},{k,0,n*(n+1)/2}]
%Y A336420 A000110 gives row sums.
%Y A336420 A000124 gives row lengths.
%Y A336420 A000142 counts divisors of superprimorials.
%Y A336420 A006939 lists superprimorials or Chernoff numbers.
%Y A336420 A008278 is the version counting only distinct prime factors.
%Y A336420 A008302 counts divisors of superprimorials by bigomega.
%Y A336420 A022915 counts permutations of prime indices of superprimorials.
%Y A336420 A076954 can be used instead of A006939.
%Y A336420 A130091 lists numbers with distinct prime multiplicities.
%Y A336420 A146291 counts divisors by bigomega.
%Y A336420 A181796 counts divisors with distinct prime multiplicities.
%Y A336420 A181818 gives products of superprimorials.
%Y A336420 A317829 counts factorizations of superprimorials.
%Y A336420 A336417 counts perfect-power divisors of superprimorials.
%Y A336420 A336498 counts divisors of factorials by bigomega.
%Y A336420 A336499 uses factorials instead superprimorials.
%Y A336420 Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.
%K A336420 nonn,tabf
%O A336420 0,5
%A A336420 _Gus Wiseman_, Jul 25 2020