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Number of divisors of n that are both composite and squarefree.
3

%I #11 Sep 28 2024 12:58:23

%S 0,0,0,0,0,1,0,0,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,1,0,1,0,1,0,4,0,0,1,1,

%T 1,1,0,1,1,1,0,4,0,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,4,0,1,1,0,1,4,0,1,

%U 1,4,0,1,0,1,1,1,1,4,0,1,0,1,0,4,1,1,1

%N Number of divisors of n that are both composite and squarefree.

%C Also number of composite and squarefree m <= n such that rad(m) | n, i.e., in row n of A162306, where rad = A007947.

%C This sequence is distinct from A327517; A327517(210) != a(210).

%C Record setters are primorials, a(6) = 1, a(30) = 4, a(210) = 11, etc., since primorials P(n) = A002110(n) are the smallest instance of omega(n) = A001221(n).

%H Michael De Vlieger, <a href="/A376504/b376504.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A376504/a376504.png">Hasse diagram of row 1440 of A162306</a> showing 4 squarefree composites in green, 3 primes in red, the empty product in gray, 17 perfect powers of primes in yellow, and 72 numbers that are neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.

%F a(n) = 2^omega(n) - omega(n) - 1 = A034444(n) - A001221(n) - 1.

%F a(n) = 0 for n = p^m, where p is prime and m >= 0, i.e., n in A000961.

%F a(n) = A000295(omega(n)) = A000295(A001221(n)).

%t Array[2^# - # - 1 &@ PrimeNu[#] &, 120]

%Y Cf. A000005, A000295, A000961, A001221, A002110, A007947, A034444, A120944, A162306, A327517, A361373 (number of prime powers in row n of A162306), A374514 (number of divisors of n that are neither squarefree nor prime powers).

%K nonn,easy

%O 1,30

%A _Michael De Vlieger_, Sep 25 2024