A350100 - OEIS

%I #21 Dec 23 2024 11:52:52

%S 2,3,5,11,23,30,41,50,76,100,149,159,189,345,437,509,693,1110,1165,

%T 5018,14908,18906,19079,28634,38682,80444,105686,185179,265236,269697,

%U 409049,558269,1673629,2965232,3528015,4292936,34919969,43957056,148793437,187220890,424171123

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

%C a(50) (in b-file) > 5*10^10, corresponding to A378904(50) > 613. - _Hugo Pfoertner_, Dec 23 2024

%H Hugo Pfoertner, <a href="/A350100/b350100.txt">Table of n, a(n) for n = 1..49</a>

%e   n  a(n)  p1   a(n)^2   p2   gap=2*A378904(n)

%e   1   2     3      4      5    2

%e   2   3     7      9     11    4

%e   3   5    23     25     29    6

%e   4  11   113    121    127   14

%e   5  23   523    529    541   18

%e   6  30   887    900    907   20

%e   7  41  1669   1681   1693   24

%e   8  50  2477   2500   2503   26

%t 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 (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(3000000)

%o (Python)

%o from itertools import count, islice

%o from sympy import prevprime, nextprime

%o def A350100_gen(): # generator of terms

%o     c = 0

%o     for k in count(2):

%o         a = nextprime(m:=k**2)-prevprime(m)

%o         if a>c:

%o             yield k

%o             c = a

%o A350100_list = list(islice(A350100_gen(),20)) # _Chai Wah Wu_, Dec 17 2024

%Y A378904 are the corresponding gaps, divided by 2.

%Y Cf. A001223, A005250, A058043.

%K nonn,changed

%O 1,1

%A _Hugo Pfoertner_, Dec 25 2021