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A376504
Number of divisors of n that are both composite and squarefree.
3
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, 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, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, 0, 1, 0, 4, 1, 1, 1
OFFSET
1,30
COMMENTS
Also number of composite and squarefree m <= n such that rad(m) | n, i.e., in row n of A162306, where rad = A007947.
This sequence is distinct from A327517; A327517(210) != a(210).
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).
LINKS
Michael De Vlieger, Hasse diagram of row 1440 of A162306 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.
FORMULA
a(n) = 2^omega(n) - omega(n) - 1 = A034444(n) - A001221(n) - 1.
a(n) = 0 for n = p^m, where p is prime and m >= 0, i.e., n in A000961.
a(n) = A000295(omega(n)) = A000295(A001221(n)).
MATHEMATICA
Array[2^# - # - 1 &@ PrimeNu[#] &, 120]
CROSSREFS
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).
Sequence in context: A152889 A216273 A327517 * A151905 A226997 A245965
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Sep 25 2024
STATUS
approved