A378904 -id:A378904 - cp's OEIS frontend

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

Hugo Pfoertner, Dec 25 2021

nonn

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

			  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

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

A378904 are the corresponding gaps, divided by 2.

Cf. A001223, A005250, A058043.

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 *)

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)

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

A379444 a(n) is the difference between the least prime > (n+1)^2 and the largest prime < n^2, divided by 2.

Original entry on oeis.org

4, 5, 8, 7, 11, 10, 11, 11, 15, 18, 17, 15, 17, 17, 21, 24, 25, 21, 23, 24, 31, 27, 30, 29, 30, 30, 40, 34, 40, 39, 35, 38, 38, 37, 41, 40, 42, 45, 48, 54, 51, 51, 47, 56, 50, 51, 57, 52, 66, 57, 60, 57, 64, 57, 65, 71, 65, 69, 67, 64, 78, 66, 68, 69, 72, 77, 81

Offset: 2

Hugo Pfoertner, Dec 23 2024

2*a(n) would be the gap needed between consecutive primes to provide a counterexample to Legendre's conjecture that there is always a prime between n^2 and (n+1)^2. The gaps actually observed are significantly smaller; see A378904 for comparison.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A001223, A007491, A053001, A058043, A350100, A378904.

a[n_]:=(NextPrime[(n+1)^2] - NextPrime[n^2,-1])/2; Array[a,67,2] (* Stefano Spezia, Jan 24 2025 *)

a379444(n) = (nextprime((n+1)^2) - precprime(n^2))/2

a(n) = (A007491(n+1) - A053001(n))/2.

a(n) >= n + 2.

cp's OEIS Frontend

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

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