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Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).
10

%I #9 Jul 27 2020 00:24:18

%S 1,3,13,76,571,5309,59341,780149

%N Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

%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(2) = 13 ways:

%e 12/1/1 12/2/1 12/3/1 12/4/1 12/12/1

%e 12/2/2 12/3/3 12/4/2 12/12/2

%e 12/4/4 12/12/3

%e 12/12/4

%e 12/12/12

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

%t strsig[n_]:=UnsameQ@@Last/@FactorInteger[n];

%t Table[Total[Cases[Divisors[chern[n]],d_?strsig:>Count[Divisors[d],e_?strsig]]],{n,0,5}]

%Y A000258 shifted once to the left is dominated by this sequence.

%Y A336422 is the generalization to non-superprimorials.

%Y A000110 counts divisors of superprimorials with distinct prime exponents.

%Y A006939 lists superprimorials or Chernoff numbers.

%Y A008302 counts divisors of superprimorials by bigomega.

%Y A022915 counts permutations of prime indices of superprimorials.

%Y A076954 can be used instead of A006939.

%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 Cf. A000005, A008278, A027423, A071625, A118914, A124010, A327498, A336417, A336419, A336420, A336426, A336500, A336568.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Jul 25 2020