A309359 -id:A309359 - cp's OEIS frontend

A309329 Median of primes with n decimal digits.

4, 47, 509, 5273, 53047, 532887, 5356259, 53765483, 539119753, 5402600081, 54118210435, 541947386821, 5425907665571, 54313871643797, 543611236251491, 5440228524355329, 54438462600610510, 544705097744731559, 5449909581264135103

Offset: 1

Hugo Pfoertner, Jul 25 2019

The number of n-digit primes < a(n) equals the number of n-digit primes > a(n). The median of an even number of values is understood to be defined as the arithmetic mean of the two central elements.

			a(1) = 4 because {2, 3, 5, 7} are the 4 one-digit primes. The 2 central elements of the sorted list are 3 and 5. 4 = (3 + 5)/2.
a(2) = 47 because it is the central element of the sorted list of the A006879(2) = 21 two-digit primes. There are 10 such primes < 47 and 10 such primes > 47.

Cf. A006879, A006880, A309359

a(n) = (prime(A006880(n-1) + ceiling(A006879(n)/2)) + prime(A006880(n-1) + floor(A006879(n)/2) + 1)) / 2.

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

A328032 If there are m primes between 10^(n-1) and 10^n, a(n) is the middle prime if m is odd, otherwise the larger of the two middle primes.

Original entry on oeis.org

5, 47, 509, 5273, 53047, 532907, 5356259, 53765519, 539119753, 5402600081, 54118210441, 541947386821, 5425907665571, 54313871643797, 543611236251491, 5440228524355381, 54438462600610513, 544705097744731559, 5449909581264135103

Offset: 1

Robert G. Wilson v, Oct 02 2019

This sequence, unlike A309329, only contains primes.
For n > 2, a(n) > 10*a(n-1) for the terms shown. Does this continue?
The prime index of a(n): 3, 15, 97, 699, 5411, 44046, 371539, 3213018, 28304495, 252950023, 2286553663, 20862983416, 191836724429, 1775503643821, 16524756086736, 154541455728298, 1451397749344080, 13681755722697547, 129398810782042734, 1227438634918631724, 11674044544289825385, 111297278087667319110, 1063393839148059937607, 10180460079478002418395, 97640954583246485139774, 938046530135790455369642, 9025853588857058793877502, ..., .

			a(1) is 5 since, among the single-digit primes, i.e., {2, 3, 5, 7}, the two middle primes are {3, 5}, of which the larger one is 5;
a(2) is 47 since it is the middle prime of the two-digit primes, i.e., {11, 13, 17, ..., 47, ..., 83, 89, 97};
a(3) is 509 since it is the middle prime of the three-digit primes, i.e., {101, 103, 107, ..., 509, ..., 983, 991, 997}.

Cf. A006879, A006880, A309329, A309359.

f[n_] := Block[{p = PrimePi[ 10^(n -1)], q = PrimePi[ 10^n]}, Prime[ Ceiling[(q +p +1)/2]]]; Array[f, 13]

a(n) is the next prime after A309329(n) - 1.

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A309329 Median of primes with n decimal digits.

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

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A328032 If there are m primes between 10^(n-1) and 10^n, a(n) is the middle prime if m is odd, otherwise the larger of the two middle primes.

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