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A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).

%I A360543 #47 Mar 14 2023 06:39:49
%S A360543 0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,6,0,0,0,0,11,0,0,
%T A360543 0,1,0,0,0,1,0,0,0,0,1,0,0,2,5,1,0,0,0,2,0,1,0,0,0,0,0,0,1,26,0,0,0,0,
%U A360543 0,0,0,4,0,0,2,0,0,0,0,3,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,3,1,4
%N A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).
%H A360543 Michael De Vlieger, <a href="/A360543/b360543.txt">Table of n, a(n) for n = 1..16384</a>
%H A360543 Michael De Vlieger, <a href="/A360543/a360543.png">Diagram showing k <= n</a>, n = 1..36, where a(n) is the number of numbers k in row n shown in blue. Numbers k in green in row n are counted by A360480(n). Together, blue and green numbers are counted by A243823(n) and appear in row n of A272619. Dots at (k, n) in red are divisors, and in yellow and magenta in row n are counted by A243822(n).
%H A360543 Michael De Vlieger, <a href="/A360543/a360543_1.png">Plot k < n at (x, y) = (k, -n)</a> for n = 1..2^10, where black represents k such that gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k).
%F A360543 a(n) = A243823(n) - A360480(n).
%F A360543 a(n) = A045763(n) - A243822(n) - A360480(n).
%F A360543 a(n) = A051953(n) - A000005(n) - A243822(n) - A360480(n).
%F A360543 a(n) = A051953(n) - A010846(n) - A360480(n).
%F A360543 a(n) = A243823(n) = A045763(n) for n in A246547.
%F A360543 For prime power n = p^e, n > 1, a(n) = p^(e-1) - e.
%F A360543 For n in A360765, a(n) > 0.
%e A360543 a(4) = 0 since k = 1..3 are prime powers.
%e A360543 a(8) = 1 since only k = 6 is such that p = 3, q = 5, but gcd(6, 10) = 2.
%e A360543 a(9) = 1 since the following satisfies definition: {6},
%e A360543 a(16) = 4, i.e., {6, 10, 12, 14},
%e A360543 a(25) = 3, i.e., {10, 15, 20},
%e A360543 a(27) = 6, i.e., {6, 12, 15, 18, 21, 24},
%e A360543 a(32) = 11, i.e., {6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30},
%e A360543 a(36) = 1, i.e., {30},
%e A360543 a(40) = 1, i.e., {30},
%e A360543 a(45) = 1, i.e., {30}, etc.
%t A360543 nn = 120; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; c = Select[Range[4, nn], CompositeQ]; Table[Function[{q, r}, Count[TakeWhile[c, # <= n &], _?(And[PrimeNu[#] > q, Divisible[rad[#], r]] &)]] @@ {PrimeNu[n], rad[n]}, {n, nn}]
%Y A360543 Cf. A000005, A000010, A001221, A007947, A010846, A045763, A051953, A243822, A243823, A246547, A272619, A360480, A360765.
%K A360543 nonn
%O A360543 1,16
%A A360543 _Michael De Vlieger_, Mar 06 2023