A014085 -id:A014085 - cp's OEIS frontend

A345755 a(n) is the number of primes p satisfying n(log_2(n))^2 < p <= (n+1)(log_2(n+1))^2.

1, 3, 2, 3, 3, 4, 4, 4, 5, 3, 4, 4, 6, 3, 5, 7, 3, 4, 6, 5, 5, 7, 5, 3, 6, 6, 7, 6, 4, 6, 5, 7, 5, 6, 5, 6, 7, 6, 8, 4, 6, 6, 9, 3, 5, 7, 9, 5, 7, 9, 4, 7, 7, 5, 7, 6, 5, 9, 7, 8, 3, 7, 8, 8, 8, 6, 4, 7, 6, 8, 10, 7, 8, 7, 6, 7, 6, 6, 6, 7, 7, 10, 4, 8, 9, 7

Offset: 1

Hal M. Switkay, Jun 27 2021

nonn

Prime gaps appear to grow more slowly than any power function.
Cramér's conjecture states that prime gaps grow as follows: prime(n+1) - prime(n) = O(log(prime(n))^2).
Since prime(n) ~ n*log(n), we conjecture that a(n) > 0 for n > 0, and that the exponent 2 cannot be replaced by any smaller exponent.
Note: n*(log_2(n))^2 < n^(log(127)/log(16)) when n >= 267. Therefore the conjecture immediately above is stronger than the conjecture that A143935(n) > 0 when n > 0, which in turn is stronger than Legendre's conjecture.
This sequence relies on intervals that are slightly more than twice as wide as those in the similar sequence A166363. A comment at that sequence by Greathouse discovers zero values (representing prime-free intervals). In contrast, the present sequence does not include zero entries for n <= 2772, suggesting that the lengths of prime gaps may be bracketed by the two sequences. We conjecture that prime gaps may be larger than log(p)^2, but are not larger than log_2(p)^2. - Hal M. Switkay, Aug 29 2023

			a(10) is the number of primes > 110.35 and <= 131.64. a(10) = 3, because the primes in this interval are 113, 127, and 131.

Hal M. Switkay, Table of n, a(n) for n = 1..2772
Wikipedia, Cramér's conjecture

Cf. A014085, A143935, A166363.

Differences @ Table[PrimePi[n*Log2[n]^2], {n, 1, 100}] (* Amiram Eldar, Jun 27 2021 *)

f(n) = n*(log(n)/log(2))^2;
a(n) = primepi(f(n+1)) - primepi(f(n)); \\ Michel Marcus, Jun 30 2021

A348194 a(n) = A077767(n) - A077766(n).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Oct 06 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014085, A077766, A077767, A091295, A348193.

a(n) = sum(k=n^2, (n+1)^2, (isprime(k)&&k%4==3)-(isprime(k)&&k%4==1));

a(n) = A348193(n+1) - A348193(n).

A349791 a(n) is the median of the primes between n^2 and (n+1)^2.

Original entry on oeis.org

6, 12, 19, 30, 42, 59, 72, 89, 107, 134, 157, 181, 205, 236, 271, 311, 348, 381, 421, 461, 503, 560, 601, 650, 701, 754, 821, 870, 933, 994, 1051, 1113, 1193, 1268, 1319, 1423, 1482, 1559, 1624, 1723, 1801, 1884, 1993, 2081, 2148, 2267, 2357, 2444, 2549, 2663

Offset: 2

Hugo Pfoertner, Dec 05 2021

nonn

The median of an even number of values is assumed to be defined as the arithmetic mean of the two central elements in their sorted list. The special case of the primes 2 and 3 in the interval [1,4] is excluded because their median would be 5/2.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A000290, A000720, A014085, A309359, A349792.

Table[Median@Select[Range[n^2,(n+1)^2],PrimeQ],{n,2,51}] (* Giorgos Kalogeropoulos, Dec 05 2021 *)

medpsq(n) = {my(p1=nextprime(n^2), p2=precprime((n+1)^2), np1=primepi(p1), np2=primepi(p2), nm=(np1+np2)/2);
if(denominator(nm)==1, prime(nm), (prime(nm-1/2)+prime(nm+1/2))/2)};
for(k=2,51,print1(medpsq(k),", "))

from sympy import primerange
from statistics import median
def a(n): return int(median(primerange(n**2, (n+1)**2)))
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Dec 05 2021

from sympy import primepi, prime
def A349791(n):
    b = primepi(n**2)+primepi((n+1)**2)+1
    return (prime(b//2)+prime((b+1)//2))//2 if b % 2 else prime(b//2) # Chai Wah Wu, Dec 05 2021

A349792 Numbers k such that k*(k+1) is the median of the primes between k^2 and (k+1)^2.

Original entry on oeis.org

2, 3, 5, 6, 8, 25, 29, 38, 59, 101, 135, 217, 260, 295, 317, 455, 551, 686, 687, 720, 825, 912, 1193, 1233, 1300, 1879, 1967, 2200, 2576, 2719, 2857, 3303, 3512, 4215, 4241, 4448, 4658, 5825, 5932, 5952, 6155, 6750, 7275, 10305, 10323, 10962, 11279, 13495, 14104

Offset: 1

Hugo Pfoertner, Dec 05 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..552 (terms 1..85 from Hugo Pfoertner)

Cf. A000290, A000720, A002378, A014085, A349791.

Select[Range@3000,Median@Select[Range[#^2,(#+1)^2],PrimeQ]==#(#+1)&] (* Giorgos Kalogeropoulos, Dec 05 2021 *)

a349791(n) = {my(p1=nextprime(n^2), p2=precprime((n+1)^2), np1=primepi(p1), np2=primepi(p2), nm=(np1+np2)/2); if(denominator(nm)==1, prime(nm), (prime(nm-1/2)+prime(nm+1/2))/2)};
for(k=2,5000, my(t=k*(k+1)); if(t==a349791(k),print1(k,", ")))

from sympy import primerange
from statistics import median
def ok(n): return n>1 and int(median(primerange(n**2, (n+1)**2)))==n*(n+1)
print([k for k in range(999) if ok(k)]) # Michael S. Branicky, Dec 05 2021

from itertools import count, islice
from sympy import primepi, prime, nextprime
def A349792gen(): # generator of terms
    p1 = 0
    for n in count(1):
        p2 = primepi((n+1)**2)
        b = p1 + p2 + 1
        if b % 2:
            p = prime(b//2)
            q = nextprime(p)
            if p+q == 2*n*(n+1):
                yield n
        p1 = p2
A349792_list = list(islice(A349792gen(),12)) # Chai Wah Wu, Dec 08 2021

A037037 Number of primes between n and 3n.

Original entry on oeis.org

2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 6, 7, 7, 8, 9, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 14, 14, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 17, 18, 18, 18, 19, 20, 19, 19, 20, 21, 21, 22, 21, 22, 23, 23, 24, 24, 24, 25, 24, 24, 25, 26, 27, 28, 27, 27, 27, 28, 27, 27, 26, 27, 28

Offset: 1

Felice Russo

nonn

			For example a(5)=4 because between 5 and 15 there are 4 primes: 5, 7, 11 and 13.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A014085.

Array[Count[Range[#, 3 #], k_ /; PrimeQ@ k] &, 76] (* Michael De Vlieger, Mar 20 2017 *)

a(n) = primepi(3*n) - primepi(n-1); \\ Michel Marcus, Sep 28 2013

a(n) = A000720(3n) - A000720(n-1). - Wesley Ivan Hurt, Jun 15 2013

More terms from Erich Friedman

A037038 Number of primes between n and 4n+1.

Original entry on oeis.org

3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 10, 12, 12, 13, 14, 14, 14, 15, 16, 16, 16, 17, 18, 20, 21, 21, 20, 20, 20, 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 26, 28, 28, 28, 29, 30, 31, 31, 31, 32, 31, 31, 32, 34, 35, 35, 36, 36, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 41

Offset: 1

Felice Russo

nonn

			a(5)=6 because between 5 and 21 there are 6 primes: 5, 7, 11, 13, 17 and 19.

Cf. A000720 (pi), A014085.

with(numtheory); A037038:=n->pi(4*n + 1) - pi(n - 1); seq(A037038(n), 1..100); # Wesley Ivan Hurt, Feb 14 2014

Table[PrimePi[4 n + 1] - PrimePi[n - 1], {n, 1, 100}] (* Wesley Ivan Hurt, Feb 14 2014 *)

a(n) = primepi(4*n+1) - primepi(n-1); \\ Michel Marcus, Sep 28 2013

a(n) = A000720(4*n+1) - A000720(n-1). - Michel Marcus, Sep 28 2013

More terms from Erich Friedman

A069160 Number of primes p such that n^2 < p < n^2 + pi(n), where pi(n) is the number of primes less than n.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Apr 09 2002

A more restrictive version of the conjecture that there is always a prime between n^2 and (n+1)^2.

			a(10)= 2 because pi(10) = 4 and there are 2 primes between 100 and 104.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A000720, A014085.

Mathematica
```
maxN=100; lst={}; For[i=1, i
				
```

A069163 Number of symmetric primes between n^2 and (n+2)^2. Two primes are termed symmetric in n^2 to (n+2)^2 if there is a k < 2n such that mid-k and mid+k are both prime, where mid =n(n+2).

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 2, 1, 2, 4, 1, 2, 3, 1, 5, 3, 1, 4, 4, 3, 5, 3, 2, 4, 4, 1, 4, 4, 2, 5, 4, 0, 6, 2, 3, 4, 4, 2, 4, 8, 0, 3, 4, 2, 5, 4, 4, 5, 5, 3, 7, 5, 3, 5, 7, 2, 4, 6, 3, 7, 7, 5, 6, 6, 5, 5, 7, 5, 6, 8, 1, 3, 8, 3, 11, 6, 1, 10, 5, 2, 5, 8, 5, 5, 7, 5, 4, 6, 2, 8, 7, 4, 13, 7, 5, 9, 7, 4, 9

Offset: 1

T. D. Noe, Apr 09 2002

This relates primes between n^2 and (n+1)^2 to primes between (n+1)^2 and (n+2)^2. It appears that the number of symmetric primes is zero for only n=0,32,41.

			a(5) = 1 because in the range 25 to 49, the primes 29 and 41 are the only primes symmetric about the number 35.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A014085.

Mathematica
```
maxN=100; lst={}; For[n=1, n
				
```

A078764 List primes between (2n)^2 and (2n+1)^2.

Original entry on oeis.org

5, 7, 17, 19, 23, 37, 41, 43, 47, 67, 71, 73, 79, 101, 103, 107, 109, 113, 149, 151, 157, 163, 167, 197, 199, 211, 223, 257, 263, 269, 271, 277, 281, 283, 331, 337, 347, 349, 353, 359, 401, 409, 419, 421, 431, 433, 439, 487, 491, 499, 503, 509, 521, 523, 577

Offset: 1

Donald S. McDonald, Jan 09 2003

The primes may be on adjacent sides of Ulam square prime-spiral. 7 8 9 / 6 1 2 / 5 4 3/...

			The 3 primes between 4^2=16 (even) and 5^2=25 (odd) are just a(3)=17, a(4)=19 and a(5)=23.

Cf. A014085, A078763.

f[n_] := Select[Range[(2n)^2, (2n + 1)^2], PrimeQ];Flatten@Array[f, 12] (* Ray Chandler, May 03 2007 *)

Extended by Ray Chandler, May 03 2007

A104481 Bisection of A104477.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 4, 3, 5, 6, 4, 5, 5, 6, 6, 6, 5, 8, 7, 6, 7, 8, 7, 7, 9, 8, 9, 8, 9, 8, 8, 11, 10, 11, 10, 8, 11, 10, 12, 9, 12, 14, 9, 10, 14, 15, 11, 12, 10, 11, 16, 12, 15, 15, 12, 16, 14, 13, 15, 14, 14, 16, 12, 20, 14, 14, 16, 16, 17, 21, 13, 17, 22, 12, 19, 18, 19

Offset: 1

Alexandre Wajnberg, Apr 18 2005

a(n) = A014085(n) - 1. - Klaus Brockhaus, Apr 20 2005

Cf. A014085, A061265, A104477.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; f[n_] := PrimePi[ PrevPrim[(n + 1)^2]] - PrimePi[ NextPrim[n^2]]; Table[ f[n], {n, 83}] (* Robert G. Wilson v, Apr 23 2005 *)

More terms from Robert G. Wilson v, Apr 23 2005

cp's OEIS Frontend

A345755 a(n) is the number of primes p satisfying n*(log_2(n))^2 < p <= (n+1)*(log_2(n+1))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A348194 a(n) = A077767(n) - A077766(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A349791 a(n) is the median of the primes between n^2 and (n+1)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

A349792 Numbers k such that k*(k+1) is the median of the primes between k^2 and (k+1)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

A037037 Number of primes between n and 3n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A037038 Number of primes between n and 4n+1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A069160 Number of primes p such that n^2 < p < n^2 + pi(n), where pi(n) is the number of primes less than n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A069163 Number of symmetric primes between n^2 and (n+2)^2. Two primes are termed symmetric in n^2 to (n+2)^2 if there is a k < 2n such that mid-k and mid+k are both prime, where mid =n(n+2).

Views

A345755 a(n) is the number of primes p satisfying n(log_2(n))^2 < p <= (n+1)(log_2(n+1))^2.