A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

1, 1, 2, 2, 6, 4, 12, 8, 20, 28, 68, 40, 80, 0, 56, 160, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Aug 08 2020

Keywords

• nonn

Comments

Does this sequence converge to zero?

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

From Edward Moody, Jan 18 2021: (Start)

a(n) = 0 for n >= 17.

Proof: 17 is the third Ramanujan prime (A104272). Therefore, for n>=17, there are at least three primes greater than n/2 and less than or equal to n. These primes must have exponent 1 in the prime factorization of n!, therefore, at least two of them must have exponent 1 in the prime factorization of either d or n!/d, so d and n!/d cannot both have distinct prime multiplicities. (End)

Examples

			The a(1) = 1 through a(7) = 8 divisors:
  1  1  2  1   3   1    5
     2  3  2   5   2    7
           3   24  5    45
           8   40  9    63
           12      16   80
           24      18   112
                   40   720
                   45   1008
                   80
                   144
                   360
                   720
		

Crossrefs

A336419 is the version for superprimorials.

A336500 is the generalization to non-factorials.

A336616 is the maximum among these divisors.

A336617 is the minimum among these divisors.

A336939 has these row sums.

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 n! with distinct prime multiplicities.

A336415 counts divisors of n! with equal prime multiplicities.

A336423 counts chains using A130091.

Cf. A098859, A118914, A124010, A327527, A336424, A336568, A336571, A336870.

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

Programs

Mathematica

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

Extensions

a(31)-a(80) from Edward Moody, Jan 19 2021