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

%I A349792 #37 Dec 13 2021 15:18:56
%S A349792 2,3,5,6,8,25,29,38,59,101,135,217,260,295,317,455,551,686,687,720,
%T A349792 825,912,1193,1233,1300,1879,1967,2200,2576,2719,2857,3303,3512,4215,
%U A349792 4241,4448,4658,5825,5932,5952,6155,6750,7275,10305,10323,10962,11279,13495,14104
%N A349792 Numbers k such that k*(k+1) is the median of the primes between k^2 and (k+1)^2.
%H A349792 Chai Wah Wu, <a href="/A349792/b349792.txt">Table of n, a(n) for n = 1..552</a> (terms 1..85 from Hugo Pfoertner)
%t A349792 Select[Range@3000,Median@Select[Range[#^2,(#+1)^2],PrimeQ]==#(#+1)&] (* _Giorgos Kalogeropoulos_, Dec 05 2021 *)
%o A349792 (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)};
%o A349792 for(k=2,5000, my(t=k*(k+1)); if(t==a349791(k),print1(k,", ")))
%o A349792 (Python)
%o A349792 from sympy import primerange
%o A349792 from statistics import median
%o A349792 def ok(n): return n>1 and int(median(primerange(n**2, (n+1)**2)))==n*(n+1)
%o A349792 print([k for k in range(999) if ok(k)]) # _Michael S. Branicky_, Dec 05 2021
%o A349792 (Python)
%o A349792 from itertools import count, islice
%o A349792 from sympy import primepi, prime, nextprime
%o A349792 def A349792gen(): # generator of terms
%o A349792     p1 = 0
%o A349792     for n in count(1):
%o A349792         p2 = primepi((n+1)**2)
%o A349792         b = p1 + p2 + 1
%o A349792         if b % 2:
%o A349792             p = prime(b//2)
%o A349792             q = nextprime(p)
%o A349792             if p+q == 2*n*(n+1):
%o A349792                 yield n
%o A349792         p1 = p2
%o A349792 A349792_list = list(islice(A349792gen(),12)) # _Chai Wah Wu_, Dec 08 2021
%Y A349792 Cf. A000290, A000720, A002378, A014085, A349791.
%K A349792 nonn
%O A349792 1,1
%A A349792 _Hugo Pfoertner_, Dec 05 2021