cp's OEIS Frontend

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.

%I A336419 #12 Aug 31 2020 19:49:50
%S A336419 1,2,4,10,24,64,184,536,1608,5104,16448,55136,187136,658624,2339648,
%T A336419 8618208,31884640,121733120,468209408,1849540416,7342849216
%N 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.
%C A336419 A number has distinct prime exponents iff its prime signature is strict.
%C A336419 The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
%e A336419 The a(0) = 1 through a(3) = 10 divisors:
%e A336419   1  2  12  360
%e A336419 -----------------
%e A336419   1  1   1    1
%e A336419      2   3    5
%e A336419          4    8
%e A336419         12    9
%e A336419              18
%e A336419              20
%e A336419              40
%e A336419              45
%e A336419              72
%e A336419             360
%t A336419 chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t A336419 Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,6}]
%o A336419 (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 A336419 a(n)={recurse(n,1,1,1)} \\ _Andrew Howroyd_, Aug 30 2020
%Y A336419 A000110 shifted once to the left dominates this sequence.
%Y A336419 A006939 lists superprimorials or Chernoff numbers.
%Y A336419 A022915 counts permutations of prime indices of superprimorials.
%Y A336419 A130091 lists numbers with distinct prime exponents.
%Y A336419 A181796 counts divisors with distinct prime exponents.
%Y A336419 A181818 gives products of superprimorials.
%Y A336419 A317829 counts factorizations of superprimorials.
%Y A336419 A336417 counts perfect-power divisors of superprimorials.
%Y A336419 Cf. A000005, A000178, A008278, A071625, A076954, A118914, A124010, A327498, A327527, A336420, A336421, A336426, A336500, A336568.
%K A336419 nonn,more
%O A336419 0,2
%A A336419 _Gus Wiseman_, Jul 25 2020
%E A336419 a(10)-a(20) from _Andrew Howroyd_, Aug 31 2020