OFFSET

0,5

COMMENTS

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.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.

EXAMPLE

Triangle begins:

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 37 25 23 15 23 10 5 2 1 1

The divisors counted in row n = 4 are:

1 2 4 8 16 48 144 432 2160 10800 75600

3 9 12 24 72 360 720 3024

5 25 18 40 80 400 1008

7 20 54 108 504 1200

27 56 112 540 2800

28 135 200 600

45 189 675 756

50 1350

63 1400

75 4725

175

MATHEMATICA

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];

Table[Length[Select[Divisors[chern[n]], PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]], {n, 0, 5}, {k, 0, n*(n+1)/2}]

CROSSREFS

A000110 gives row sums.

A000124 gives row lengths.

A000142 counts divisors of superprimorials.

A006939 lists superprimorials or Chernoff numbers.

A008278 is the version counting only distinct prime factors.

A008302 counts divisors of superprimorials by bigomega.

A022915 counts permutations of prime indices of superprimorials.

A130091 lists numbers with distinct prime multiplicities.

A146291 counts divisors by bigomega.

A181796 counts divisors with distinct prime multiplicities.

A181818 gives products of superprimorials.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

A336498 counts divisors of factorials by bigomega.

A336499 uses factorials instead superprimorials.

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jul 25 2020

STATUS

approved