A376846 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, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 0, 3, 1, 1, 0, 4, 0, 2, 0, 3, 0, 0, 0, 3, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 0, 3, 0, 0, 0, 7, 0, 3, 1, 5, 0, 1, 0, 4, 0, 3, 0, 8, 0, 1, 0, 4, 0, 4, 0, 4, 2

Offset: 1

Author

Michael De Vlieger, Oct 06 2024

Keywords

• nonn

Comments

Number of m not exceeding n such that the squarefree kernel of m divides n, and m has more prime factors with repetition than does n.

Number of m in row n of A162306 such that Omega(m) > Omega(n).

Examples

			Table of select n such that a(n) > 0:
   n  a(n)  List of m such that Omega(m) > Omega(n).
  -------------------------------------------------
  10   1    {8}
  14   1    {8}
  18   1    {16}
  20   1    {16}
  22   2    {8, 16}
  26   2    {8, 16}
  28   1    {16}
  30   2    {16, 24}
  33   1    {27}
  34   3    {8, 16, 32}
  36   1    {32}
  38   3    {8, 16, 32}
  39   1    {27}
  40   1    {32}
  42   4    {16, 24, 32, 36}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Hasse diagrams of m in select rows n of A162306 indicating in red those m such that Omega(m) > Omega(n).
• Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024, about zeros in this sequence.

Crossrefs

Cf. A000961, A001222, A007947, A010846, A162306.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
{0}~Join~Table[With[{k = PrimeOmega[n]}, Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)]], {n, 2, 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) < A010846(n).