A075084 Number of composite numbers c with n <= c <= 2*n.

0, 1, 2, 3, 4, 5, 5, 7, 7, 7, 8, 9, 10, 12, 12, 12, 13, 15, 15, 17, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 26, 27, 27, 28, 28, 28, 30, 31, 31, 32, 33, 34, 36, 36, 37, 38, 40, 40, 41, 41, 41, 42, 43, 43, 44, 44, 45, 46, 48, 49, 51, 52, 52, 53, 53, 54, 56, 56, 56, 57, 59, 60

Offset

1,3

Comments

The number of composite numbers <= n is n less the number of primes less 1.

The sequence is nondecreasing.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Formula

a(n) = n - pi(2n) + pi(n-1) + 1, for n>1.

Example

a(8) = 7: the composite numbers are 8,9,10,12,14,15 and 16.

Maple

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A075084 := proc(n) chi(2*n) - chi(n-1); end;
a := [seq(A075084(n), n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Mathematica

Table[n - PrimePi[2n] + PrimePi[n - 1] + 1, {n, 2, 75}]

Prog

(Python)
from sympy import primepi
def A075084(n): return n+primepi(n-1)-primepi(n<<1)+1 if n>1 else 0 # Chai Wah Wu, Oct 20 2024
(PARI) a(n) = if (n>1, n - primepi(2*n) + primepi(n-1) + 1, 0); \\ Michel Marcus, Oct 21 2024

Crossrefs

Related sequences:

Primes (p) and composites (c): A000040, A002808, A000720, A065855.

Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.

Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

Sequence in context: A181894 A265535 A094802 * A086593 A073757 A088468

Adjacent sequences: A075081 A075082 A075083 * A075085 A075086 A075087

Keyword

easy,nonn

Author

Amarnath Murthy, Sep 11 2002

Extensions

Edited by Robert G. Wilson v, Sep 12 2002

Definition clarified by N. J. A. Sloane, Oct 20 2024

Status

approved