A082997 a(n) = card{ x <= n : omega(x) = 2 }.
0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 19, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36, 36, 36, 37, 37, 38, 39, 40, 41
Offset: 1
Keywords
References
- G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
Links
- Daniel Suteu, Table of n, a(n) for n = 1..10000
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+`if`(nops(ifactors(n)[2])=2, 1, 0)) end: seq(a(n), n=1..100); # Alois P. Heinz, Aug 23 2021 -
Mathematica
a[n_] := Count[PrimeNu[Range[n]], 2]; Array[a, 100] (* Jean-François Alcover, Mar 02 2022 *)
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PARI
a(n)=sum(i=1,n,if(omega(i)-2,0,1))
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PARI
a(n) = my(s = sqrtint(n), p = 2, j = 1, count = 0); while(p <= s, my(r = nextprime(p+1)); my(t = p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021
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Python
from sympy import factorint from itertools import accumulate def cond(n): return int(len(factorint(n))==2) def aupto(nn): return list(accumulate(map(cond, range(1, nn+1)))) print(aupto(77)) # Michael S. Branicky, Jul 21 2021
Formula
a(n) ~ (n/log(n))*log(log(n)).
a(A007774(n)) = n. - Daniel Suteu, Jul 21 2021