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

2, 3, 5, 11, 23, 30, 41, 50, 76, 100, 149, 159, 189, 345, 437, 509, 693, 1110, 1165, 5018, 14908, 18906, 19079, 28634, 38682, 80444, 105686, 185179, 265236, 269697, 409049, 558269, 1673629, 2965232, 3528015, 4292936, 34919969, 43957056, 148793437, 187220890, 424171123

Offset

1,1

Comments

a(51) (in b-file) > 1.5*10^11, corresponding to A378904(51) > 723. - Hugo Pfoertner, Jan 04 2025

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..50

Example

  n  a(n)  p1   a(n)^2   p2   gap=2*A378904(n)
  1   2     3      4      5    2
  2   3     7      9     11    4
  3   5    23     25     29    6
  4  11   113    121    127   14
  5  23   523    529    541   18
  6  30   887    900    907   20
  7  41  1669   1681   1693   24
  8  50  2477   2500   2503   26

Mathematica

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 *)

Prog

(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))};
a350100(3000000)
(Python)
from itertools import count, islice
from sympy import prevprime, nextprime
def A350100_gen(): # generator of terms
    c = 0
    for k in count(2):
        a = nextprime(m:=k**2)-prevprime(m)
        if a>c:
            yield k
            c = a
A350100_list = list(islice(A350100_gen(), 20)) # Chai Wah Wu, Dec 17 2024

Crossrefs

A378904 are the corresponding gaps, divided by 2.

Cf. A001223, A005250, A058043.

Sequence in context: A118571 A118504 A215714 * A038905 A019405 A079848

Adjacent sequences: A350097 A350098 A350099 * A350101 A350102 A350103

Keyword

nonn

Author

Hugo Pfoertner, Dec 25 2021

Status

approved