A336417 Number of perfect-power divisors of superprimorials A006939.

1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522

Offset: 0

Author

Gus Wiseman, Jul 24 2020

Keywords

• nonn

Comments

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

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

Examples

			The a(0) = 1 through a(4) = 15 divisors:
  1  2  12  360  75600
-------------------------
  1  1   1    1      1
         4    4      4
              8      8
              9      9
             36     16
                    25
                    27
                    36
                   100
                   144
                   216
                   225
                   400
                   900
                  3600
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

A000325 is the uniform version.

A076954 can be used instead of A006939.

A336416 gives the same for factorials instead of superprimorials.

A000217 counts prime power divisors of superprimorials.

A000961 gives prime powers.

A001597 gives perfect powers, with complement A007916.

A006939 gives superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A091050 counts perfect power divisors.

A181818 gives products of superprimorials.

A294068 counts factorizations using perfect powers.

A317829 counts factorizations of superprimorials.

Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.

Programs

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]

PARI

a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020

Formula

a(n) = A091050(A006939(n)).

a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 30 2020