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

1,2

Comments

In a counterexample to the Legendre conjecture, 2*a(n) > A350100(n) would have to hold; i.e., the primes adjacent to a square k^2 would have to have a difference > 2k+1. Therefore, a(50) would have to be larger by an order of magnitude of 10^9. - Hugo Pfoertner, Dec 27 2025

Links

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

Plot of a(n) vs log(A350100), using Plot 2.

Prog

(PARI) 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)
(Python)
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

Crossrefs

Cf. A058043, A350100.

Sequence in context: A140221 A046668 A047533 * A060525 A152863 A047359

Adjacent sequences: A378901 A378902 A378903 * A378905 A378906 A378907

Keyword

nonn,more

Author

Hugo Pfoertner, Dec 15 2024

Extensions

a(50) from Hugo Pfoertner, Jan 04 2025

Status

approved