A065855 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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

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.

FORMULA

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

EXAMPLE

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

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 *)

PROG

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

(Haskell)

a065855 n = a065855_list !! (n-1)

a065855_list = scanl1 (+) (map a066247 [1..])

-- Reinhard Zumkeller, Oct 20 2014

(Python)

from sympy import primepi

def A065855(n):

return 0 if n < 4 else n - primepi(n) - 1 # Chai Wah Wu, Apr 14 2016