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

%I A367098 #13 Jan 08 2024 01:40:26
%S A367098 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,
%T A367098 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,
%U A367098 1,3,0,6,0,1,2,2,1,3,0,4,0,1,0,5,1,1,1
%N A367098 Number of divisors of n with exactly two distinct prime factors.
%H A367098 Amiram Eldar, <a href="/A367098/b367098.txt">Table of n, a(n) for n = 1..10000</a>
%F A367098 a(n) = (A001222(n)^2 - A090885(n))/2. - _Amiram Eldar_, Jan 08 2024
%e A367098 The a(n) divisors for n = 1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360:
%e A367098   .  6  6   6   6   6   6   6   6    6    6    6
%e A367098         12  12  12  10  12  10  12   12   12   10
%e A367098             24  18  12  18  12  18   18   18   12
%e A367098                 36  15  24  15  24   24   24   15
%e A367098                     20  36  20  36   36   36   18
%e A367098                         72  24  48   54   48   20
%e A367098                             40  72   72   72   24
%e A367098                                 144  108  96   36
%e A367098                                      216  144  40
%e A367098                                           288  45
%e A367098                                                72
%t A367098 Table[Length[Select[Divisors[n], PrimeNu[#]==2&]],{n,100}]
%t A367098 a[1] = 0; a[n_] := (Total[(e = FactorInteger[n][[;; , 2]])]^2 - Total[e^2])/2; Array[a, 100] (* _Amiram Eldar_, Jan 08 2024 *)
%o A367098 (PARI) a(n) = {my(e = factor(n)[, 2]); (vecsum(e)^2 - e~*e)/2;} \\ _Amiram Eldar_, Jan 08 2024
%Y A367098 For just one distinct prime factor we have A001222 (prime-power divisors).
%Y A367098 This sequence counts divisors belonging to A007774.
%Y A367098 Counting all prime factors gives A086971, firsts A220264.
%Y A367098 Column k = 2 of A146289.
%Y A367098 - Positions of zeros are A000961 (powers of primes), complement A024619.
%Y A367098 - Positions of ones are A006881 (squarefree semiprimes).
%Y A367098 - Positions of twos are A054753.
%Y A367098 - Positions of first appearances are A367099.
%Y A367098 A001221 counts distinct prime factors.
%Y A367098 A001358 lists semiprimes, complement A100959.
%Y A367098 A367096 lists semiprime divisors, sum A076290.
%Y A367098 Cf. A000005, A001248, A056170, A079275, A090885, A366740.
%K A367098 nonn,easy
%O A367098 1,12
%A A367098 _Gus Wiseman_, Nov 09 2023