A361235 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).

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, 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, 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

Offset: 1

Author

Michael De Vlieger, Mar 06 2023

Keywords

• nonn

Comments

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

Examples

			a(6) = 1 since k = 4 is such that rad(4)|rad(6) = 2|6 and omega(4) < omega(6).
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).
a(12) = 2 since the following satisfies definition: {8, 9}.
a(14) = 2, i.e., {4, 8}.
a(15) = 1, i.e., {9}.
a(18) = 3, i.e., {8, 9, 16}.
a(30) = 10, i.e., {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Diagram showing k <= n, 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).
• Michael De Vlieger, Plot k < n at (x, y) = (k, -n) 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).

Crossrefs

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

Programs

Mathematica

nn = 2^10;
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
{0}~Join~Table[
   If[PrimePowerQ[n], 0,
    q = PrimeNu[n]; r = rad[n];
    Count[ DeleteCases[ Range[n],
     _?(Or[Divisible[n, #], CoprimeQ[#, n], ! Divisible[r, rad[#]]] &)],
     _?(PrimeNu[#] < q &)]],
   {n, 2, nn}]

Formula

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

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

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

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

a(n) = 0 for n in A000961.

a(n) > 0 for n in A013929.

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