A350100 -id:A350100 - cp's OEIS frontend

A378904 2*a(n) are the gaps that correspond to A350100(n).

1, 2, 3, 7, 9, 10, 12, 13, 15, 17, 18, 20, 26, 27, 29, 33, 39, 41, 66, 75, 84, 90, 95, 100, 113, 126, 140, 144, 155, 162, 177, 204, 206, 210, 216, 302, 303, 364, 389, 391, 399, 418, 441, 469, 492, 497, 504, 520, 613, 723

Offset: 1

Hugo Pfoertner, Dec 15 2024

Cf. A058043, A350100.

a378904(kmax) = my(d=0); for(k=2, kmax, my(k2=k*k, dd=(nextprime(k2)-precprime(k2))/2); if(dd>d, print1(dd,", "); d=dd));
a378904(10^6)

from itertools import count, islice
from sympy import prevprime, nextprime
def A378904_gen(): # generator of terms
    c = 0
    for k in count(2):
        a = nextprime(m:=k**2)-prevprime(m)
        if a>c:
            yield a>>1
            c = a
A378904_list = list(islice(A378904_gen(),20)) # Chai Wah Wu, Dec 17 2024

a(50) from Hugo Pfoertner, Jan 04 2025

A378429 Numbers k such that the prime gap between the consecutive primes p1 < k# = primorial(k) < p2 sets a new record.

Original entry on oeis.org

3, 7, 13, 17, 23, 29, 37, 43, 47, 61, 71, 79, 83, 97, 101, 109, 137, 193, 347, 349, 409, 457, 587, 599, 887, 929, 967, 1319, 1801, 1877, 2081, 2687, 2731, 2741, 2843, 2939, 2957, 3673, 3823, 4621, 5717, 6011, 6151, 6563, 6863, 7393, 8389, 9833, 11903, 12547

Offset: 1

Jean-Marc Rebert, Dec 20 2024

nonn

			a(1) = 3, because the prime gap between the consecutive primes 5 < 3# < 7 sets the first record of 2.
 n gap       p1        a(n)#        p2
 1   2             5 <   3# <             7
 2  12           199 <   7# <           211
 3  18         30029 <  13# <         30047
 4  48        510481 <  17# <        510529
 5  80     223092827 <  23# <     223092907
 6 102    6469693189 <  29# <    6469693291
 7 120 7420738134751 <  37# < 7420738134871

Cf. A001223, A058044, A034386, A350100.

a(32)-a(39) from Amiram Eldar, Dec 20 2024

a(40)-a(50) from Michael S. Branicky, Dec 21 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.

A379449 Numbers k such that the prime gap between the consecutive primes p1 < k! = factorial(k) < p2 sets a new record.

Original entry on oeis.org

3, 4, 5, 8, 13, 19, 24, 29, 34, 45, 47, 51, 56, 61, 71, 107, 127, 140, 184, 192, 198, 274, 284, 375, 384, 559, 592, 630, 689, 774, 792, 834, 1133, 1213, 1241, 1315, 1947

Offset: 1

Jean-Marc Rebert, Dec 23 2024

			a(1) = 3, because the prime gap between the consecutive primes 5 < 3! < 7 sets the first record of 2.
 n gap            p1            <  x! <             p2
 1   2                        5 <  3! <                        7;
 2   6                       23 <  4! <                       29;
 3  14                      113 <  5! <                      127;
 4  54                    40289 <  8! <                    40343;
 5  90               6227020777 < 13! <               6227020867;
 6 190       121645100408831899 < 19! <       121645100408832089;
 7 204 620448401733239439359927 < 24! < 620448401733239439360131;

Cf. A054588, A350100, A378429.

a(37) from Jinyuan Wang, Dec 23 2024

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A378904 2*a(n) are the gaps that correspond to A350100(n).

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A378429 Numbers k such that the prime gap between the consecutive primes p1 < k# = primorial(k) < p2 sets a new record.

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

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A379449 Numbers k such that the prime gap between the consecutive primes p1 < k! = factorial(k) < p2 sets a new record.

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