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A309179 Primes to which a record size square needs to be added to reach another prime. 0
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 (list; graph; refs; listen; history; text; internal format)
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 A331399 A084599

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

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)