%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