A307912 a(n) = n - 1 - pi(2*n-1) + pi(n), where pi is the prime counting function.

0, 0, 1, 1, 3, 3, 4, 5, 5, 5, 7, 7, 9, 10, 10, 10, 12, 13, 14, 15, 15, 15, 17, 17, 18, 19, 19, 20, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 31, 31, 33, 34, 34, 35, 37, 38, 38, 39, 39, 39, 41, 41, 41, 42, 42, 43, 45, 46, 48, 49, 50, 50, 51, 51, 53, 54

Offset

1,5

Comments

For n > 1, a(n) is the number of composites in the closed interval [n+1, 2n-1].

a(n) is also the number of composites appearing among the largest parts of the partitions of 2n into two distinct parts.

Links

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

Index entries for sequences related to partitions

Formula

a(n) = n - 1 - A060715(n).

a(n) = n - 1 - A000720(2*n-1) + A000720(n).

Example

a(7) = 4; there are 4 composites in the closed interval [8, 13]: 8, 9, 10 and 12.

Maple

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A307912 := proc(n) chi(2*n-1) - chi(n); end;
A := [seq(A307912(n), n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Mathematica

Table[n - 1 - PrimePi[2 n - 1] + PrimePi[n], {n, 100}]

Prog

(Python)
from sympy import primepi
def A307912(n): return n+primepi(n)-primepi((n<<1)-1)-1 # Chai Wah Wu, Oct 20 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: A072648 A185585 A072945 * A274004 A196592 A120180

Adjacent sequences: A307909 A307910 A307911 * A307913 A307914 A307915

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 09 2019

Status

approved