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.

%I A381096 #4 Feb 16 2025 23:02:47
%S A381096 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,
%T A381096 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,
%U A381096 20,29,0,42,0,31,25,26,16,45,0,34,24,45,0,42,0,37
%N A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.
%C A381096 Number of k <= n in the cototient of n that do not share the same squarefree kernel as n.
%C A381096 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.
%C A381096 Define quality Q(k) to be true if k is such that 1 < gcd(k,n) and rad(k) != rad(n).
%C A381096 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).
%H A381096 Michael De Vlieger, <a href="/A381096/b381096.txt">Table of n, a(n) for n = 1..10000</a>
%H A381096 Michael De Vlieger, <a href="/A381096/a381096.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%F A381096 a(1) = 0, a(p) = a(4) = 0.
%F A381096 a(n) = A045763(n) - A005361(n).
%F A381096 For n > 1, a(n) = n - phi(n) - tau(n/rad(n)) = A000010(n) - A005361(n).
%F A381096 For n > 1, a(n) = n - A000010(n) - A008479(n) + A355432(n).
%e A381096 a(6) = 3 since {2, 3, 4} are neither coprime to 6 and do not have the squarefree kernel 6.
%e A381096 a(8) = 1 since only 6 is neither coprime to 8 and does not share the squarefree kernel 2 with 8.
%e A381096 a(10) = 5 since {2, 4, 5, 6, 8} are neither coprime to 10 nor have the squarefree kernel 10.
%e A381096 a(12) = 6 since {2, 3, 4, 8, 9, 10} are neither coprime to 12 nor have the squarefree kernel 6.
%e A381096 a(14) = 7 since {2, 4, 6, 7, 8, 10, 12} are neither coprime to 14 nor have the squarefree kernel 14, etc.
%t A381096 {0}~Join~Table[n - EulerPhi[n] - DivisorSigma[0, n/rad[n]], {n, 2, 120}]
%Y A381096 Cf. A000010, A005361, A008479, A355432, A359929, A381094.
%K A381096 nonn,easy
%O A381096 1,6
%A A381096 _Michael De Vlieger_, Feb 14 2025