

A065855


Number of composites <= n.


46



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
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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 YaPing Lu and ShuFang Deng (2020), Table 3.1.  N. J. A. Sloane, Mar 26 2020


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
YaPing Lu and ShuFang 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.


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[nPrimePi[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) { for (n=1, 1000, a=n  primepi(n)  1; write("b065855.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009
(Haskell)
a065855 n = a065855_list !! (n1)
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


CROSSREFS

Cf. A000720, A062298, A002808.
Cf. A066247.
Sequence in context: A057062 A283993 A255572 * A236863 A242976 A218445
Adjacent sequences: A065852 A065853 A065854 * A065856 A065857 A065858


KEYWORD

easy,nonn,nice


AUTHOR

Labos Elemer, Nov 26 2001


STATUS

approved



