A367098 Number of divisors of n with exactly two distinct prime factors.

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

Offset: 1

Author

Gus Wiseman, Nov 09 2023

Keywords

• nonn
• easy

Examples

			The a(n) divisors for n = 1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360:
  .  6  6   6   6   6   6   6   6    6    6    6
        12  12  12  10  12  10  12   12   12   10
            24  18  12  18  12  18   18   18   12
                36  15  24  15  24   24   24   15
                    20  36  20  36   36   36   18
                        72  24  48   54   48   20
                            40  72   72   72   24
                                144  108  96   36
                                     216  144  40
                                          288  45
                                               72
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

For just one distinct prime factor we have A001222 (prime-power divisors).

This sequence counts divisors belonging to A007774.

Counting all prime factors gives A086971, firsts A220264.

Column k = 2 of A146289.

- Positions of zeros are A000961 (powers of primes), complement A024619.

- Positions of ones are A006881 (squarefree semiprimes).

- Positions of twos are A054753.

- Positions of first appearances are A367099.

A001221 counts distinct prime factors.

A001358 lists semiprimes, complement A100959.

A367096 lists semiprime divisors, sum A076290.

Cf. A000005, A001248, A056170, A079275, A090885, A366740.

Programs

Mathematica

Table[Length[Select[Divisors[n], PrimeNu[#]==2&]],{n,100}]
a[1] = 0; a[n_] := (Total[(e = FactorInteger[n][[;; , 2]])]^2 - Total[e^2])/2; Array[a, 100] (* Amiram Eldar, Jan 08 2024 *)

PARI

a(n) = {my(e = factor(n)[, 2]); (vecsum(e)^2 - e~*e)/2;} \\ Amiram Eldar, Jan 08 2024

Formula

a(n) = (A001222(n)^2 - A090885(n))/2. - Amiram Eldar, Jan 08 2024