A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.

0, 0, 0, 0, 0, 3, 0, 1, 1, 5, 0, 6, 0, 7, 6, 4, 0, 10, 0, 10, 8, 11, 0, 13, 3, 13, 6, 14, 0, 21, 0, 11, 12, 17, 10, 20, 0, 19, 14, 21, 0, 29, 0, 22, 19, 23, 0, 28, 5, 28, 18, 26, 0, 33, 14, 29, 20, 29, 0, 42, 0, 31, 25, 26, 16, 45, 0, 34, 24, 45, 0, 42, 0, 37

Offset

1,6

Comments

Number of k <= n in the cototient of n that do not share the same squarefree kernel as n.

Define a number k "neutral" to n to be such that 1 < gcd(k,n) < k, that is, k neither divides n nor is coprime to n. A045763(n) is the number of k < n such that k is neutral to n.

Define quality Q(k) to be true if k is such that 1 < gcd(k,n) and rad(k) != rad(n).

Then for k <= n and n > 1, a(n) = A045763(n), but admitting divisors k | n such that rad(k) != rad(n), and eliminating occasional nondivisors k such that rad(k) = rad(n), i.e., k listed in row n of A359929 for n = A360768(i).

Links

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

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Formula

a(1) = 0, a(p) = a(4) = 0.

a(n) = A045763(n) - A005361(n).

For n > 1, a(n) = n - phi(n) - tau(n/rad(n)) = A000010(n) - A005361(n).

For n > 1, a(n) = n - A000010(n) - A008479(n) + A355432(n).

Example

a(6) = 3 since {2, 3, 4} are neither coprime to 6 and do not have the squarefree kernel 6.
a(8) = 1 since only 6 is neither coprime to 8 and does not share the squarefree kernel 2 with 8.
a(10) = 5 since {2, 4, 5, 6, 8} are neither coprime to 10 nor have the squarefree kernel 10.
a(12) = 6 since {2, 3, 4, 8, 9, 10} are neither coprime to 12 nor have the squarefree kernel 6.
a(14) = 7 since {2, 4, 6, 7, 8, 10, 12} are neither coprime to 14 nor have the squarefree kernel 14, etc.

Mathematica

{0}~Join~Table[n - EulerPhi[n] - DivisorSigma[0, n/rad[n]], {n, 2, 120}]

Crossrefs

Cf. A000010, A005361, A008479, A355432, A359929, A381094.

Sequence in context: A094921 A395047 A246522 * A140166 A242782 A011256

Adjacent sequences: A381093 A381094 A381095 * A381097 A381098 A381099

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 14 2025

Status

approved