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

Author

Michael De Vlieger, Sep 25 2024

Keywords

• nonn
• easy

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.

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).

Programs

Mathematica

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

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)).