A334655 Number of integers less than n with the same number of distinct prime factors as n.

0, 0, 1, 2, 3, 0, 4, 5, 6, 1, 7, 2, 8, 3, 4, 9, 10, 5, 11, 6, 7, 8, 12, 9, 13, 10, 14, 11, 15, 0, 16, 17, 12, 13, 14, 15, 18, 16, 17, 18, 19, 1, 20, 19, 20, 21, 21, 22, 22, 23, 24, 25, 23, 26, 27, 28, 29, 30, 24, 2, 25, 31, 32, 26, 33, 3, 27, 34, 35, 4, 28, 36, 29, 37, 38, 39, 40, 5, 30, 41

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = |{j < n : omega(j) = omega(n)}|.

a(n) = A067003(n) - 1.

Example

a(12) = 2 because omega(12) = 2 and also omega(6) = omega(10) = 2.

Maple

R:= NULL:
for n from 1 to 100 do
  w:= nops(numtheory:-factorset(n));
  if assigned(V[w]) then V[w]:= V[w]+1 else V[w]:= 1 fi;
  R:= R, V[w]-1
od:
R; # Robert Israel, Feb 25 2024

Mathematica

Table[Length[Select[Range[n - 1], PrimeNu[#] == PrimeNu[n] &]], {n, 80}]

Prog

(PARI) a(n)={my(t=omega(n)); sum(k=1, n-1, omega(k)==t)} \\ Andrew Howroyd, Oct 31 2020

Crossrefs

Cf. A001221, A002110 (positions of 0's), A047983, A067003, A067004, A322837, A322841, A335097.

Sequence in context: A193331 A091246 A271439 * A343410 A133637 A258093

Adjacent sequences: A334652 A334653 A334654 * A334656 A334657 A334658

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 31 2020

Status

approved