A336939 Irregular triangle read by rows where T(n,k) is the number of divisors d of n! with k prime factors (counting multiplicity), such that both d and n!/d have distinct prime multiplicities.

1, 1, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 0, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 4, 2, 2, 2, 2, 4, 0, 2, 0, 0, 2, 0, 4, 0, 4, 4, 4, 4, 0, 4, 0, 2, 0, 1, 3, 2, 6, 4, 5, 7, 6, 6, 7, 5, 4, 6, 2, 3, 1, 0, 2, 0, 4, 2, 2, 4, 4, 4, 4, 4, 2, 2, 4, 0, 2, 0

Offset

0,6

Comments

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

Links

Table of n, a(n) for n=0..93.

Example

Triangle begins:
  1
  1
  1 1
  0 2 0
  1 2 0 2 1
  0 2 0 0 2 0
  1 2 1 2 2 1 2 1
  0 2 0 2 0 2 0 2 0
  0 2 0 4 2 2 2 2 4 0 2 0
  0 2 0 4 0 4 4 4 4 0 4 0 2 0
  1 3 2 6 4 5 7 6 6 7 5 4 6 2 3 1
Row n = 8 counts the following divisors (empty columns shown as dots):
  .  5  .  20  40   80  360   720   640  .  5760  .
     7     28  56  112  504  1008   896     8064
           45                      1440
           63                      2016

Mathematica

Table[Length[Select[Divisors[n!], UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n!/#]&&PrimeOmega[#]==k&]], {n, 0, 10}, {k, 0, PrimeOmega[n!]}]

Crossrefs

A022559 gives row lengths minus one.

A336500 is the generalization to all positive integers.

A336868 gives the first (also last) column.

A336869 gives row sums.

A336870 is the version for superprimorials.

A000005 counts divisors.

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.

A336414 counts divisors of factorials with distinct prime multiplicities.

A336415 counts divisors of factorials with equal prime multiplicities.

A336423 counts chains using A130091.

Cf. A006939, A098859, A118914, A124010, A336419, A336424, A336571, A336865.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336414, A336415, A336416.

Sequence in context: A053603 A371222 A085794 * A239002 A004548 A125921

Adjacent sequences: A336936 A336937 A336938 * A336940 A336941 A336942

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 08 2020

Status

approved