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Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.
15

%I #12 Aug 31 2020 19:49:50

%S 1,2,4,10,24,64,184,536,1608,5104,16448,55136,187136,658624,2339648,

%T 8618208,31884640,121733120,468209408,1849540416,7342849216

%N Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.

%C A number has distinct prime exponents iff its prime signature is strict.

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

%e The a(0) = 1 through a(3) = 10 divisors:

%e 1 2 12 360

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

%e 1 1 1 1

%e 2 3 5

%e 4 8

%e 12 9

%e 18

%e 20

%e 40

%e 45

%e 72

%e 360

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

%t Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,6}]

%o (PARI) recurse(n,k,b,d)={if(k>n, 1, sum(i=0, k, if((i==0||!bittest(b,i)) && (i==k||!bittest(d,k-i)), self()(n, k+1, bitor(b, 1<<i), bitor(d, 1<<(k-i))))))}

%o a(n)={recurse(n,1,1,1)} \\ _Andrew Howroyd_, Aug 30 2020

%Y A000110 shifted once to the left dominates this sequence.

%Y A006939 lists superprimorials or Chernoff numbers.

%Y A022915 counts permutations of prime indices of superprimorials.

%Y A130091 lists numbers with distinct prime exponents.

%Y A181796 counts divisors with distinct prime exponents.

%Y A181818 gives products of superprimorials.

%Y A317829 counts factorizations of superprimorials.

%Y A336417 counts perfect-power divisors of superprimorials.

%Y Cf. A000005, A000178, A008278, A071625, A076954, A118914, A124010, A327498, A327527, A336420, A336421, A336426, A336500, A336568.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Jul 25 2020

%E a(10)-a(20) from _Andrew Howroyd_, Aug 31 2020