login
Number of perfect-power divisors of superprimorials A006939.
15

%I #14 Aug 30 2020 20:20:43

%S 1,1,2,5,15,44,169,652,3106,15286,89933,532476,3698650,25749335,

%T 204947216,1636097441,14693641859,132055603656,1319433514898,

%U 13186485900967,144978145009105,1594375302986404,19128405558986057,229508085926717076,2983342885319348522

%N Number of perfect-power divisors of superprimorials A006939.

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

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

%H Andrew Howroyd, <a href="/A336417/b336417.txt">Table of n, a(n) for n = 0..200</a>

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

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

%e The a(0) = 1 through a(4) = 15 divisors:

%e 1 2 12 360 75600

%e -------------------------

%e 1 1 1 1 1

%e 4 4 4

%e 8 8

%e 9 9

%e 36 16

%e 25

%e 27

%e 36

%e 100

%e 144

%e 216

%e 225

%e 400

%e 900

%e 3600

%t chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];

%t perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];

%t Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]

%o (PARI) a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ _Andrew Howroyd_, Aug 30 2020

%Y A000325 is the uniform version.

%Y A076954 can be used instead of A006939.

%Y A336416 gives the same for factorials instead of superprimorials.

%Y A000217 counts prime power divisors of superprimorials.

%Y A000961 gives prime powers.

%Y A001597 gives perfect powers, with complement A007916.

%Y A006939 gives superprimorials or Chernoff numbers.

%Y A022915 counts permutations of prime indices of superprimorials.

%Y A091050 counts perfect power divisors.

%Y A181818 gives products of superprimorials.

%Y A294068 counts factorizations using perfect powers.

%Y A317829 counts factorizations of superprimorials.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 24 2020

%E Terms a(10) and beyond from _Andrew Howroyd_, Aug 30 2020