A359929 -id:A359929 - cp's OEIS frontend

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

Michael De Vlieger, Feb 14 2025

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).

			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.

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.

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

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

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).

A359382 a(n) = number of k < t such that rad(k) = rad(t) and k does not divide t, where t = A360768(n) and rad(k) = A007947(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 2, 1, 1, 1, 4, 2, 2, 4, 1, 1, 1, 1, 3, 1, 3, 2, 8, 1, 2, 1, 7, 2, 1, 2, 5, 2, 1, 1, 3, 3, 1, 6, 1, 1, 5, 5, 4, 5, 1, 1, 4, 8, 3, 3, 1, 2, 1, 4, 2, 3, 5, 10, 2, 1, 3, 3, 1, 1, 1, 6, 1, 3, 7, 1, 1, 7, 3, 14, 3, 6, 3, 2, 1, 1, 2, 7, 2, 1, 1, 2, 2, 8, 4, 6, 4, 8, 1, 1, 2, 1, 6, 9, 2, 1

Offset: 1

Michael De Vlieger, Mar 29 2023

nonn

This sequence contains nonzero values in A355432.

			Table relating a(n) to b(n) = A360768(n) and row n of A359929.
n  b(n)   row n of A359929   a(n)
---------------------------------
1   18    12                   1
2   24    18                   1
3   36    24                   1
4   48    18, 36               2
5   50    20, 40               2
6   54    12, 24, 36, 48       4

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, highlighting records (A360768(n) in A360589) in red.

Cf. A007947, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A355432, A359929, A360589, A360768.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[671], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
t = Select[s, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@
      {#, FactorInteger[#][[All, 1]]} &];
Map[Function[{n, k},
    Count[TakeWhile[s, # < n &],
      _?(And[rad[#] == k, ! Divisible[n, #]] &)]] @@ {#, rad[#]} &, t]

a(n) = A355432(A360768(n)) = length of row n in A359929.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula