A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 4, 2, 2, 2, 5, 2, 4, 2, 4, 2, 2, 2, 6, 3, 2, 4, 4, 2, 0, 2, 6, 2, 2, 2, 6, 2, 2, 2, 6, 2, 0, 2, 4, 4, 2, 2, 8, 3, 4, 2, 4, 2, 6, 2, 6, 2, 2, 2, 4, 2, 2, 4, 7, 2, 0, 2, 4, 2, 0, 2, 8, 2, 2, 4, 4, 2, 0, 2, 8, 5, 2, 2, 4, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Aug 06 2020

Keywords

• nonn

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Examples

			The a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16
		

Crossrefs

A336419 is the version for superprimorials.

A336568 gives positions of zeros.

A336869 is the restriction to factorials.

A007425 counts divisors of divisors.

A056924 counts divisors greater than their quotient.

A074206 counts chains of divisors from n to 1.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime multiplicities.

A336424 counts factorizations using A130091.

A336422 counts divisible pairs of divisors, both in A130091.

A327498 gives the maximum divisor with distinct prime multiplicities.

A336423 counts chains in A130091, with maximal version A336569.

A336568 gives numbers not a product of two elements of A130091.

A336571 counts divisor sets using A130091, with maximal version A336570.

Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Programs

Mathematica

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]