A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 0, 3, 0, 4, 0, 0, 1, 1, 1, 2, 0, 1, 1, 5, 0, 5, 0, 3, 2, 1, 0, 6, 0, 1, 1, 3, 0, 1, 1, 5, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 3, 1, 5, 0, 4, 0, 1, 1, 3, 1, 5, 0, 8, 0, 1, 0, 11, 1, 1, 1

Offset: 1

Author

Michael De Vlieger, Oct 13 2024

Keywords

• nonn

Examples

			Table of select n such that a(n) > 0:
   n  a(n)  List of m in A376248 such that Omega(m) <= Omega(n)
  -------------------------------------------------------------
   6    1   {9}
  10    1   {25}
  12    2   {18, 27}
  14    1   {49}
  15    1   {25}
  18    1   {27}
  20    3   {25, 50, 125}
  24    4   {27, 36, 54, 81}
  28    3   {49, 98, 343}
  30    4   {45, 50, 75, 125}
  40    5   {50, 100, 125, 250, 625}
  48    6   {54, 72, 81, 108, 162, 243}
  60   11   {75, 81, 90, 100, 125, 135, 150, 225, 250, 375, 625}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Hasse diagrams of m in select R(n), where R(n) is the union of rows n of A162306 and A376248, indicating in blue those m > n such that Omega(m) <= Omega(n).

Crossrefs

Cf. A000961, A001221, A001222, A007947, A162306, A376567, A376846, A068795.

Programs

Maple

with(NumberTheory):
cond := (m, n) -> irem(n, Radical(m)) = 0 and Omega(m) <= Omega(n):
a := n -> nops(select(m -> cond(m, n), [seq(n+1..A068795(n))])):
seq(a(n), n = 1..87);  # Peter Luschny, Oct 25 2024

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; w = PrimeNu[n]; Binomial[k + w, w] - Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)], {n, 120}]

Formula

a(n) = card({m > n : rad(m) | n, Omega(m) <= Omega(n) }).

a(n) = 0 for prime power n (in A000961).

a(n) = card(A376248 \ A162306).

a(n) = A376567(n) - A010846(n) + A376546(n) = binomial(A001222(n) + A001221(n), A001221(n)) - A010846(n) + A376546(n).