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.

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, 1, 6, 5, 29, 23, 33, 87, 70, 78, 74, 129, 84, 81, 49, 39, 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

Offset: 0

Author

Gus Wiseman, Jul 25 2020

Keywords

• nonn
• tabf

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.

Examples

			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
		

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.

A076954 can be used instead of A006939.

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.

Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.

Programs

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}]