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%I #17 Sep 02 2020 23:04:03
%S 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 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 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 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 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 The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
%C 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 Triangle begins:
%e 1
%e 1 1
%e 1 2 1 1
%e 1 3 2 5 2 1 1
%e 1 4 3 11 7 7 10 5 2 1 1
%e 1 5 4 19 14 18 37 25 23 15 23 10 5 2 1 1
%e The divisors counted in row n = 4 are:
%e 1 2 4 8 16 48 144 432 2160 10800 75600
%e 3 9 12 24 72 360 720 3024
%e 5 25 18 40 80 400 1008
%e 7 20 54 108 504 1200
%e 27 56 112 540 2800
%e 28 135 200 600
%e 45 189 675 756
%e 50 1350
%e 63 1400
%e 75 4725
%e 175
%t chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t Table[Length[Select[Divisors[chern[n]],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,5},{k,0,n*(n+1)/2}]
%Y A000110 gives row sums.
%Y A000124 gives row lengths.
%Y A000142 counts divisors of superprimorials.
%Y A006939 lists superprimorials or Chernoff numbers.
%Y A008278 is the version counting only distinct prime factors.
%Y A008302 counts divisors of superprimorials by bigomega.
%Y A022915 counts permutations of prime indices of superprimorials.
%Y A076954 can be used instead of A006939.
%Y A130091 lists numbers with distinct prime multiplicities.
%Y A146291 counts divisors by bigomega.
%Y A181796 counts divisors with distinct prime multiplicities.
%Y A181818 gives products of superprimorials.
%Y A317829 counts factorizations of superprimorials.
%Y A336417 counts perfect-power divisors of superprimorials.
%Y A336498 counts divisors of factorials by bigomega.
%Y A336499 uses factorials instead superprimorials.
%Y Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.
%K nonn,tabf
%O 0,5
%A _Gus Wiseman_, Jul 25 2020
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