A336419 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.

1, 2, 4, 10, 24, 64, 184, 536, 1608, 5104, 16448, 55136, 187136, 658624, 2339648, 8618208, 31884640, 121733120, 468209408, 1849540416, 7342849216

Offset: 0

Author

Gus Wiseman, Jul 25 2020

Keywords

• nonn
• more

Comments

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

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

Examples

			The a(0) = 1 through a(3) = 10 divisors:
  1  2  12  360
-----------------
  1  1   1    1
     2   3    5
         4    8
        12    9
             18
             20
             40
             45
             72
            360
		

Crossrefs

A000110 shifted once to the left dominates this sequence.

A006939 lists superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime exponents.

A181818 gives products of superprimorials.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

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

Programs

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,6}]

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<Andrew Howroyd, Aug 30 2020

Extensions

a(10)-a(20) from Andrew Howroyd, Aug 31 2020