A336870 Irregular triangle read by rows where T(n,k) is the number of divisors d of the superprimorial A006939(n) with k prime factors (counting multiplicity), such that d and A006939(n)/d both have distinct prime multiplicities.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 4, 2, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 7, 7, 7, 7, 7, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 18, 10, 10, 15, 21, 21, 15, 10, 10, 18, 7, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 18, 23, 15, 20, 37, 35, 40, 46, 32, 46, 40, 35, 37, 20, 15, 23, 18, 7, 4, 4, 1, 1, 1

Offset

0,11

Comments

Are there any zeros (cf. A336939)?

A number has distinct prime multiplicities iff its prime signature is strict.

Links

Table of n, a(n) for n=0..91.

Example

Triangle begins:
  1
  1 1
  1 1 1 1
  1 1 1 4 1 1 1
  1 1 1 4 4 2 4 4 1 1 1
  1 1 1 4 4 7 7 7 7 7 7 4 4 1 1 1
Row n = 4 counts the following divisors:
  1  7  25   27   16  112   400   432  3024  10800  75600
             63   54  675  1350  1008
             75   56       1400  1200
            175  189       4725  2800

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];
Table[Length[Select[Divisors[chern[n]], UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&&PrimeOmega[#]==k&]], {n, 0, 6}, {k, 0, PrimeOmega[chern[n]]}]

Crossrefs

A000124 gives row lengths.

A336419 gives row sums.

A336500 is the generalization to all positive integers.

A336939 is the version for factorials.

A000005 counts divisors.

A000110 counts divisors of superprimorials with distinct prime multiplicities.

A000142 lists factorials.

A000325 counts divisors of superprimorials with equal prime multiplicities.

A006939 lists superprimorials.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A327498 gives the maximum divisor of n with distinct prime multiplicities.

A336423 counts chains using A130091.

Cf. A022559, A027423, A048742, A071626, A098859, A118914, A124010, A336414, A336424, A336568, A336571, A336865, A336869.

Sequence in context: A366145 A204160 A360165 * A184101 A347176 A164561

Adjacent sequences: A336867 A336868 A336869 * A336871 A336872 A336873

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 08 2020

Status

approved