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

1,4

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.

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 - A035250(n).

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

Example

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.

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}]

Prog

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

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.

Sequence in context: A162684 A265533 A145340 * A160680 A243069 A342496

Adjacent sequences: A307986 A307987 A307988 * A307990 A307991 A307992

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 09 2019

Status

approved