A062298 Number of nonprimes <= n.

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

Offset

1,4

Comments

a(n) = n - A000720(n). This is asymptotic to n - Li(n). Note that a(n) + A095117(n) = 2*n. - Jonathan Vos Post, Nov 22 2004

Same as number of primes between n and prime(n+1) and between n and prime(n)+1 (end points excluded); n prime -> a(n)=a(n-1), n composite-> a(n)=1+a(n-1). - David James Sycamore, Jul 23 2018

There exists at least one prime number between a(n) and n for n >= 3 (see the paper by Ya-Ping Lu attached in the links). - Ya-Ping Lu, Nov 27 2020

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Formula

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

a(n) = 1 + A065855(n). - David James Sycamore, Jul 23 2018

Example

a(19) = 11 as there are 8 primes up to 19 (inclusive).

Maple

NumComposites := proc(N::posint) local count, i:count := 0:for i from 1 to N do if not isprime(i) then count := count + 1 fi:od: count; end:seq(NumComposites(binomial(k+1, k)), k=0..73); # Zerinvary Lajos, May 26 2008
A062298 := proc(n) n-numtheory[pi](n) ; end: seq(A062298(n), n=1..120) ; # R. J. Mathar, Sep 27 2009

Mathematica

Table[n-PrimePi[n], {n, 80}] (* Harvey P. Dale, May 10 2012 *)
Accumulate[Table[If[PrimeQ[n], 0, 1], {n, 100}]] (* Harvey P. Dale, Feb 15 2017 *)

Prog

(PARI) a(n) = n-primepi(n); \\ Harry J. Smith, Aug 04 2009
(Haskell)
a062298 n = a062298_list !! (n-1)
a062298_list = scanl1 (+) $ map (1 -) a010051_list
-- Reinhard Zumkeller, Oct 10 2013
(Magma) [n - #PrimesUpTo(n): n in [1..100]]; // Vincenzo Librandi, Aug 05 2015
(Python)
from sympy import primepi
print([n - primepi(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 29 2017

Crossrefs

Cf. A000720, A101203, A010051, A065855.

Sequence in context: A321695 A197432 A255573 * A283371 A116579 A156253

Adjacent sequences: A062295 A062296 A062297 * A062299 A062300 A062301

Keyword

nonn

Author

Amarnath Murthy, Jun 19 2001

Extensions

Corrected and extended by Vladeta Jovovic, Jun 22 2001

Status

approved