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

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

Author

Hugo Pfoertner, Dec 05 2021

Keywords

• nonn

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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,", ")))

Python

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

Python

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