A307989 a(n) = n - pi(2*n) + pi(n-1), where pi is the prime counting function.

0, 0, 1, 2, 3, 4, 4, 6, 6, 6, 7, 8, 9, 11, 11, 11, 12, 14, 14, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 23, 25, 26, 26, 27, 27, 27, 29, 30, 30, 31, 32, 33, 35, 35, 36, 37, 39, 39, 40, 40, 40, 41, 42, 42, 43, 43, 44, 45, 47, 48, 50, 51, 51, 52, 52, 53, 55

Offset: 1

Author

Wesley Ivan Hurt, May 09 2019

Keywords

• nonn
• easy

Comments

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

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

Examples

			a(7) = 4; There are 7 partitions of 2*7 = 14 into two parts (13,1), (12,2), (11,3), (10,4), (9,5), (8,6), (7,7). Among the largest parts 12, 10, 9 and 8 are composite, so a(7) = 4.
		

Links

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

Crossrefs

Cf. A000720, A035250.

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
A307989 := proc(n) chi(2*n-1) - chi(n-1); end;
a := [seq(A307989(n),n=1..120)];

Mathematica

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

Python

from sympy import primepi
def A307989(n): return n+primepi(n-1)-primepi(n<<1) # Chai Wah Wu, Oct 20 2024

Formula

a(n) = n - A035250(n).

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