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

Author

Wesley Ivan Hurt, May 09 2019

Keywords

• nonn
• easy

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.

Examples

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to partitions

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.

Programs

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}]

Python

from sympy import primepi
def A307912(n): return n+primepi(n)-primepi((n<<1)-1)-1 # Chai Wah Wu, Oct 20 2024

Formula

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

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