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A367098
Number of divisors of n with exactly two distinct prime factors.
3
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
1,12
LINKS
FORMULA
a(n) = (A001222(n)^2 - A090885(n))/2. - Amiram Eldar, Jan 08 2024
EXAMPLE
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
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 *)
PROG
(PARI) a(n) = {my(e = factor(n)[, 2]); (vecsum(e)^2 - e~*e)/2; } \\ Amiram Eldar, Jan 08 2024
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.
Sequence in context: A327695 A345446 A354911 * A343660 A319058 A281116
KEYWORD
nonn,easy
AUTHOR
Gus Wiseman, Nov 09 2023
STATUS
approved