A335097 Number of integers less than n with the same number of prime factors (counted with multiplicity) as n.

0, 0, 1, 0, 2, 1, 3, 0, 2, 3, 4, 1, 5, 4, 5, 0, 6, 2, 7, 3, 6, 7, 8, 1, 8, 9, 4, 5, 9, 6, 10, 0, 10, 11, 12, 2, 11, 13, 14, 3, 12, 7, 13, 8, 9, 15, 14, 1, 16, 10, 17, 11, 15, 4, 18, 5, 19, 20, 16, 6, 17, 21, 12, 0, 22, 13, 18, 14, 23, 15, 19, 2, 20, 24, 16, 17, 25, 18, 21, 3

Offset

1,5

Links

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

Formula

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

a(n) = A058933(n) - 1.

Example

a(10) = 3 because bigomega(10) = 2 and also bigomega(4) = bigomega(6) = bigomega(9) = 2.

Maple

A:= NULL:
for n from 1 to 100 do
  t:= numtheory:-bigomega(n);
  if not assigned(R[t]) then
    A:= A, 0;
    R[t]:= 1;
   else
    A:= A, R[t];
    R[t]:= R[t]+1;
   fi
od:
A; # Robert Israel, Oct 24 2021

Mathematica

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

Prog

(PARI) a(n)={my(t=bigomega(n)); sum(k=1, n-1, bigomega(k)==t)} \\ Andrew Howroyd, Oct 31 2020
(Python)
from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A335097(n):
    if n==1: return 0
    if isprime(n): return primepi(n)-1
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, primeomega(n)))-1) # Chai Wah Wu, Aug 28 2024

Crossrefs

Cf. A000079 (positions of 0's), A001222, A047983, A058933, A067004, A322838, A334655.

Sequence in context: A035157 A318995 A217176 * A334312 A087469 A022328

Adjacent sequences: A335094 A335095 A335096 * A335098 A335099 A335100

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 31 2020

Status

approved