OFFSET
1,10
COMMENTS
a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.
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).
FORMULA
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Mar 06 2023
STATUS
approved