A065855 Number of composites <= n.

0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 18, 19, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 45, 46, 47, 47, 48, 49, 50, 50, 51, 51, 52, 53

Offset: 1

Author

Labos Elemer, Nov 26 2001

Keywords

• easy
• nonn
• nice

Comments

Also number of primes between prime(n) and n. - Joseph L. Pe, Sep 24 2002

Plot the points (n,a(n)) by, say, appending the line ListPlot[%, PlotJoined -> True] to the Mathematica program. The result is virtually a straight line passing through the origin. For the first thousand points, the slope is approximately = 3/4. (This behavior can be explained by using the prime number theorem.) - Joseph L. Pe, Sep 24 2002

Partial sums of A066247, the characteristic function of composites. - Reinhard Zumkeller, Oct 14 2014

Appears to be the same as the coefficient h*1 of the h* polynomial for polytope representing the number n. See Ya-Ping Lu and Shu-Fang Deng (2020), Table 3.1. - _N. J. A. Sloane, Mar 26 2020

Examples

			Prime(8) = 19 and there are 3 primes between 8 and 19 (endpoints are excluded), namely 11, 13, 17. Hence a(8) = 3.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from T. D. Noe]
• Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020.

Crossrefs

Cf. A000720, A062298, A002808.

Cf. A066247.

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

Haskell

a065855 n = a065855_list !! (n-1)
a065855_list = scanl1 (+) (map a066247 [1..])
-- Reinhard Zumkeller, Oct 20 2014

Maple

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

Mathematica

(*gives number of primes < n*) f[n_] := Module[{r, i}, r = 0; i = 1; While[Prime[i] < n, i++ ]; i - 1]; (*gives number of primes between m and n with endpoints excluded*) g[m_, n_] := Module[{r}, r = f[m] - f[n]; If[PrimeQ[n], r = r - 1]; r]; Table[g[Prime[n], n], {n, 1, 1000}]
Table[n-PrimePi[n]-1,{n,75}] (* Harvey P. Dale, Jun 14 2011 *)
Accumulate[Table[If[CompositeQ[n],1,0],{n,100}]] (* Harvey P. Dale, Sep 24 2016 *)

PARI

a(n) = { n - primepi(n) - 1 } \\ Harry J. Smith, Nov 01 2009

Python

from sympy import primepi
def A065855(n):
    return 0 if n < 4 else n - primepi(n) - 1 # Chai Wah Wu, Apr 14 2016

Formula

a(n) = n - A000720(n) - 1 = A062298(n) - 1.