A309179 Primes to which a record size square needs to be added to reach another prime.

2, 3, 5, 29, 41, 389, 479, 881, 1931, 3461, 3701, 7589, 9749, 26171, 153089, 405701, 1036829, 1354349, 1516829, 2677289, 4790309, 4990961, 34648631, 46214321, 50583209, 98999969, 305094851, 331498961, 362822099, 4373372351, 11037674441, 12239355719, 16085541359

Offset

1,1

Comments

a(1) = 2 and r(1) = 1.

For n > 1, a(n) is the smallest prime for which r(n) > r(n-1) exists so that a(n) + r(n)^2 is prime and a(n) + k^2 are composite for 0 < k < r(n).

When omitting the squares in the description, the sequence becomes A002386.

Links

Table of n, a(n) for n=1..33.

Example

a(1) = 2; r(1) = 1.
a(2) = 3; 3 + 1^2 is composite, but 3 + 2^2 is prime, so r(2) = 2.
a(3) = 5; 5 + k^2 is composite for 0 < k < 6, but 5 + 6^2 is prime, so r(3) = 6.
The following are primes: 7 + 2^2, 11 + 6^2, 13 + 2^2, 17 + 6^2, 19 + 2^2, 23 + 6^2.
a(4) = 29; 29 + k^2 is composite for 0 < k < 12, but 29 + 12^2 is prime: r(4) = 12.

Prog

(Python)
from sympy import isprime, nextprime
n, p, r = 0, 0, 0
while(True):
....p = nextprime(p) ; k = 1
....while not isprime(p + k**2):
........k += 1
....if k > r:
........n += 1 ; r = k
........print("a({}) = {}".format(n, p))
(PARI) f(n) = {k=1; while(!isprime(n+k^2), k++); k; }
lista(NN) = {m=0; forprime(p=1, NN, if(f(p)>m, m=f(p); print1(p, ", ")))} \\ Jinyuan Wang, Jul 15 2019

Crossrefs

Cf. A002386, A020495, A065376, A127356, A129314.

Sequence in context: A215307 A215103 A038962 * A019400 A354084 A331399

Adjacent sequences: A309176 A309177 A309178 * A309180 A309181 A309182

Keyword

nonn

Author

Bert Dobbelaere, Jul 15 2019

Extensions

a(30)-a(33) from Giovanni Resta, Jul 16 2019

Status

approved