A376504 Number of divisors of n that are both composite and squarefree.

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

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

Adjacent sequences: A376496 A376497 A376503 * A376505 A376506 A376507

Keyword

nonn,easy,new

Author

Michael De Vlieger, Sep 25 2024

Status

approved