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A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.

%I A350100 #25 Jan 04 2025 16:10:55
%S A350100 2,3,5,11,23,30,41,50,76,100,149,159,189,345,437,509,693,1110,1165,
%T A350100 5018,14908,18906,19079,28634,38682,80444,105686,185179,265236,269697,
%U A350100 409049,558269,1673629,2965232,3528015,4292936,34919969,43957056,148793437,187220890,424171123
%N A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.
%C A350100 a(51) (in b-file) > 1.5*10^11, corresponding to A378904(51) > 723. - _Hugo Pfoertner_, Jan 04 2025
%H A350100 Hugo Pfoertner, <a href="/A350100/b350100.txt">Table of n, a(n) for n = 1..50</a>
%e A350100   n  a(n)  p1   a(n)^2   p2   gap=2*A378904(n)
%e A350100   1   2     3      4      5    2
%e A350100   2   3     7      9     11    4
%e A350100   3   5    23     25     29    6
%e A350100   4  11   113    121    127   14
%e A350100   5  23   523    529    541   18
%e A350100   6  30   887    900    907   20
%e A350100   7  41  1669   1681   1693   24
%e A350100   8  50  2477   2500   2503   26
%t A350100 Module[{nn=4242*10^5,pg},pg=Table[{n,NextPrime[n^2]-NextPrime[n^2,-1]},{n,2,nn}];DeleteDuplicates[pg,GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* _Harvey P. Dale_, Jan 28 2023 *)
%o A350100 (PARI) a350100(limit) = {my(pmax=0); for(k=2,limit, my(kk=k*k, pp=precprime(kk), pn=nextprime(kk), d=pn-pp); if(d>pmax, print1(k,", "); pmax=d))};
%o A350100 a350100(3000000)
%o A350100 (Python)
%o A350100 from itertools import count, islice
%o A350100 from sympy import prevprime, nextprime
%o A350100 def A350100_gen(): # generator of terms
%o A350100     c = 0
%o A350100     for k in count(2):
%o A350100         a = nextprime(m:=k**2)-prevprime(m)
%o A350100         if a>c:
%o A350100             yield k
%o A350100             c = a
%o A350100 A350100_list = list(islice(A350100_gen(),20)) # _Chai Wah Wu_, Dec 17 2024
%Y A350100 A378904 are the corresponding gaps, divided by 2.
%Y A350100 Cf. A001223, A005250, A058043.
%K A350100 nonn
%O A350100 1,1
%A A350100 _Hugo Pfoertner_, Dec 25 2021