A337788 -id:A337788 - cp's OEIS frontend

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

Amarnath Murthy, Jun 19 2001

nonn

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

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

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

Cf. A000720, A101203, A010051, A065855.

a062298 n = a062298_list !! (n-1)
a062298_list = scanl1 (+) $ map (1 -) a010051_list
-- Reinhard Zumkeller, Oct 10 2013

[n - #PrimesUpTo(n): n in [1..100]]; // Vincenzo Librandi, Aug 05 2015

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

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 *)

a(n) = n-primepi(n); \\ Harry J. Smith, Aug 04 2009

from sympy import primepi
print([n - primepi(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 29 2017

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

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

Corrected and extended by Vladeta Jovovic, Jun 22 2001

A095117 a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91

Offset: 0

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

Positions of first occurrences of n in A165634: A165634(a(n))=n for n>0. - Reinhard Zumkeller, Sep 23 2009

There exists at least one prime number p such that n < p <= a(n) for n >= 2. For example, 2 is in (2, 3], 5 in (3, 5], 5 in (4, 6], ..., and primes 73, 79, 83 and 89 are in (71, 91] (see Corollary 1 in the paper by Ya-Ping Lu attached in the links section). - Ya-Ping Lu, Feb 21 2021

Carmine Suriano, Table of n, a(n) for n = 0..9999
Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Complement of A095116.

Cf. A000720, A064427.

a095117 n = a000720 n + toInteger n  -- Reinhard Zumkeller, Apr 17 2012

with(numtheory): seq(n+pi(n),n=1..90); # Emeric Deutsch, May 02 2007

Table[ PrimePi@n + n, {n, 0, 71}] (* Robert G. Wilson v, Apr 22 2007 *)

a(n) = n + primepi(n); \\ Michel Marcus, Feb 21 2021

from sympy import primepi
def a(n): return primepi(n) + n
print([a(n) for n in range(72)]) # Michael S. Branicky, Feb 21 2021

a(0) = 0; for n>0, a(n) = a(n-1) + (if n is prime then 2, else 1). - Robert G. Wilson v, Apr 22 2007; corrected by David James Sycamore, Aug 16 2018

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A338521 The number of primes between n-primepi(n) and n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Ya-Ping Lu, Nov 01 2020

nonn

There is at least one prime number between n-primepi(n) and n, or a(n) >= 1, for n >= 3 (see Corollary 3 in the paper by Ya_Ping Lu attached in the links).

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

Cf. A000720, A062298, A337788.

Array[Subtract @@ Map[PrimePi, {#1 - 1, #1 - #2}] & @@ {#, PrimePi[#]} &, 105] (* Michael De Vlieger, Nov 05 2020 *)

a(n) = primepi(n - 1) - primepi(n - primepi(n)); \\ Michel Marcus, Nov 01 2020

from sympy import primepi
for n in range(1, 101):
    pi = primepi(n)
    pi_1 = primepi(n - 1)
    a = pi_1 - primepi(n - pi)
    print(a)

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

A339085 Number of primes p with n - pi(n) < p <= n + pi(n), where pi(n) is the number of primes <= n.

Original entry on oeis.org

0, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Ya-Ping Lu, Nov 23 2020

nonn

a(n) >= 2 if n >= 2 and a(n) >= 3 if n is a prime >= 3 (see the paper by Ya-Ping Lu attached in the links).

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

Cf. A000720, A095117, A062298, A337788, A338521.

Table[PrimePi[n+PrimePi[n]]-PrimePi[n-PrimePi[n]],{n,85}] (* Stefano Spezia, Nov 24 2020 *)

from sympy import primepi
for n in range(1, 101):
    m = primepi(n)
    print (primepi(n + m) - primepi(n - m))

a(n) = pi(n + pi(n)) - pi(n - pi(n)).
a(n) = A000720(n + A000720(n)) - A000720(n - A000720(n)).
a(n) = A000720(A095117(n)) - A000720(A062298(n)).
a(n) = A337788(n) + A338521(n) + isprime(n), where isprime(n) = 1 (if n is a prime) or 0 (if n is not a prime).

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5

Offset: 1

Ya-Ping Lu, Jun 24 2021

nonn

Conjecture: a(n) >= 1.

Cf. A071538, A079629, A080356, A237840, A332086, A337788.

from sympy import isprime
def istwin(m): return 1 if isprime(6*m-1)*isprime(6*m+1) == 1 else 0
ct1 = 0
for n in range(1, 100):
    ct1 += istwin(n); ct = 0
    for m in range (n + 1, n + ct1 + 1): ct += istwin(m)
    print(ct)

cp's OEIS Frontend

A062298 Number of nonprimes <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A095117 a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A338521 The number of primes between n-primepi(n) and n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A339085 Number of primes p with n - pi(n) < p <= n + pi(n), where pi(n) is the number of primes <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A344117 Number of twin prime pairs in the range (6*n + 1, 6*(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.