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a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).
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%I #9 Mar 14 2023 04:16:42

%S 0,0,0,0,0,1,0,0,0,2,0,2,0,2,1,0,0,3,0,2,1,3,0,2,0,3,0,2,0,10,0,0,2,4,

%T 1,4,0,4,2,3,0,11,0,3,2,4,0,3,0,4,2,3,0,4,1,3,2,4,0,14,0,4,2,0,1,14,0,

%U 4,2,12,0,4,0,5,2,4,1,15,0,3,0,5,0,16,1,5,3,3,0,19,1,4,3,5,1,4,0,5

%N a(n) = number of k < n, such that k does not divide n, omega(k) < omega(n) and rad(k) | rad(n), where omega(n) = A001221(n) and rad(n) = A007947(n).

%C a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.

%H Michael De Vlieger, <a href="/A361235/b361235.txt">Table of n, a(n) for n = 1..16384</a>

%H Michael De Vlieger, <a href="/A361235/a361235.png">Diagram showing k <= n</a>, n = 1..36, where a(n) is the number of numbers k in row n shown in gold. Numbers k in magenta in row n are counted by A355432(n). Together, gold and magenta numbers are counted by A243822(n) and appear in row n of A272618. Dots at (k, n) in red are divisors, and in green and blue in row n are counted by A243823(n).

%H Michael De Vlieger, <a href="/A361235/a361235_1.png">Plot k < n at (x, y) = (k, -n)</a> for n = 1..2^10, where black represents k such that k mod n != 0, such that omega(k) < omega(n) and rad(k) | rad(n).

%F a(n) = A243822(n) - A355432(n).

%F a(n) = A045763(n) - A243823(n) - A355432(n).

%F a(n) = A051953(n) - A000005(n) - A243823(n) - A355432(n) + 1.

%F a(n) = A010846(n) - A000005(n) - A355432(n).

%F a(n) = 0 for n in A000961.

%F a(n) > 0 for n in A013929.

%F a(n) = A243822(n) for n not in A360768.

%e a(6) = 1 since k = 4 is such that rad(4)|rad(6) = 2|6 and omega(4) < omega(6).

%e a(10) = 2 since k = 4 is such that rad(4)|rad(10) = 2|10 and omega(4) < omega(10), and k = 8 is such that rad(8)|rad(10) = 2|10 and omega(8) < omega(10).

%e a(12) = 2 since the following satisfies definition: {8, 9}.

%e a(14) = 2, i.e., {4, 8}.

%e a(15) = 1, i.e., {9}.

%e a(18) = 3, i.e., {8, 9, 16}.

%e a(30) = 10, i.e., {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}, etc.

%t nn = 2^10;

%t rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];

%t {0}~Join~Table[

%t If[PrimePowerQ[n], 0,

%t q = PrimeNu[n]; r = rad[n];

%t Count[ DeleteCases[ Range[n],

%t _?(Or[Divisible[n, #], CoprimeQ[#, n], ! Divisible[r, rad[#]]] &)],

%t _?(PrimeNu[#] < q &)]],

%t {n, 2, nn}]

%Y Cf. A000961, A001221, A007947, A010846, A013929, A045763, A051953, A243822, A243823, A272618, A355432.

%K nonn

%O 1,10

%A _Michael De Vlieger_, Mar 06 2023