A334655 - OEIS

%I #43 Feb 26 2024 01:28:28

%S 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,

%T 16,17,12,13,14,15,18,16,17,18,19,1,20,19,20,21,21,22,22,23,24,25,23,

%U 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

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

%H Robert Israel, <a href="/A334655/b334655.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%p R:= NULL:

%p for n from 1 to 100 do

%p   w:= nops(numtheory:-factorset(n));

%p   if assigned(V[w]) then V[w]:= V[w]+1 else V[w]:= 1 fi;

%p   R:= R, V[w]-1

%p od:

%p R; # _Robert Israel_, Feb 25 2024

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

%o (PARI) a(n)={my(t=omega(n)); sum(k=1, n-1, omega(k)==t)} \\ _Andrew Howroyd_, Oct 31 2020

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

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Oct 31 2020