cp's OEIS Frontend

A336417 Number of perfect-power divisors of superprimorials A006939.

%I A336417 #14 Aug 30 2020 20:20:43
%S A336417 1,1,2,5,15,44,169,652,3106,15286,89933,532476,3698650,25749335,
%T A336417 204947216,1636097441,14693641859,132055603656,1319433514898,
%U A336417 13186485900967,144978145009105,1594375302986404,19128405558986057,229508085926717076,2983342885319348522
%N A336417 Number of perfect-power divisors of superprimorials A006939.
%C A336417 A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.
%C A336417 The n-th superprimorial number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
%H A336417 Andrew Howroyd, <a href="/A336417/b336417.txt">Table of n, a(n) for n = 0..200</a>
%F A336417 a(n) = A091050(A006939(n)).
%F A336417 a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - _Andrew Howroyd_, Aug 30 2020
%e A336417 The a(0) = 1 through a(4) = 15 divisors:
%e A336417   1  2  12  360  75600
%e A336417 -------------------------
%e A336417   1  1   1    1      1
%e A336417          4    4      4
%e A336417               8      8
%e A336417               9      9
%e A336417              36     16
%e A336417                     25
%e A336417                     27
%e A336417                     36
%e A336417                    100
%e A336417                    144
%e A336417                    216
%e A336417                    225
%e A336417                    400
%e A336417                    900
%e A336417                   3600
%t A336417 chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t A336417 perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
%t A336417 Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]
%o A336417 (PARI) a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ _Andrew Howroyd_, Aug 30 2020
%Y A336417 A000325 is the uniform version.
%Y A336417 A076954 can be used instead of A006939.
%Y A336417 A336416 gives the same for factorials instead of superprimorials.
%Y A336417 A000217 counts prime power divisors of superprimorials.
%Y A336417 A000961 gives prime powers.
%Y A336417 A001597 gives perfect powers, with complement A007916.
%Y A336417 A006939 gives superprimorials or Chernoff numbers.
%Y A336417 A022915 counts permutations of prime indices of superprimorials.
%Y A336417 A091050 counts perfect power divisors.
%Y A336417 A181818 gives products of superprimorials.
%Y A336417 A294068 counts factorizations using perfect powers.
%Y A336417 A317829 counts factorizations of superprimorials.
%Y A336417 Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.
%K A336417 nonn
%O A336417 0,3
%A A336417 _Gus Wiseman_, Jul 24 2020
%E A336417 Terms a(10) and beyond from _Andrew Howroyd_, Aug 30 2020