A360821 Number of primes of the form k^2+1 between n^2 and 2*n^2 exclusive.

0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6

Offset

1,10

Comments

This sequence studies the special case of Bertrand's postulate where n is a square and p of the form k^2+1.

Conjecture: Except for n = 7, between the square of an integer n > 1 and the double of its square there is always at least one prime number of the form k^2+1.

The following table shows the first 15 cases, with p prime of the form k^2+1.

+----+-------+-------+-----------------------------+------+

| n | n^2 | 2*n^2 | p such that n^2 < p < 2*n^2 | a(n) |

+----+-------+-------+-----------------------------+------+

| 1 | 1 | 2 | --------------------------- | 0 |

| 2 | 4 | 8 | 5 | 1 |

| 3 | 9 | 18 | 17 | 1 |

| 4 | 16 | 32 | 17 | 1 |

| 5 | 25 | 50 | 37 | 1 |

| 6 | 36 | 72 | 37 | 1 |

| 7 | 49 | 98 | --------------------------- | 0 |

| 8 | 64 | 128 | 101 | 1 |

| 9 | 81 | 162 | 101 | 1 |

| 10 | 100 | 200 | 101, 197 | 2 |

| 11 | 121 | 242 | 197 | 1 |

| 12 | 144 | 288 | 197, 257 | 2 |

| 13 | 169 | 338 | 197, 257 | 2 |

| 14 | 196 | 392 | 197, 257 | 2 |

| 15 | 225 | 450 | 257, 401 | 2 |

Links

Michel Lagneau, Table of n, a(n) for n = 1..1000

Example

a(20) = 3 because there are 3 prime numbers of the form k^2+1 between 20^2 and 2*20^2. We obtain 400 < 401, 577, 677 < 800.

Maple

nn:=90:
for n
from 1 to nn do:
  n1:=n^2:n2:=2*n^2:it:=0:
   for p from n1+1 to n2-1 do:
    p1:=sqrt(p-1):p2:=floor(p1):
     if isprime(p) and p1=p2
      then
       it:=it+1:
       else
      fi:
    od:
printf(`%d, `, it):
od:

Prog

(PARI) a(n) = my(nb=0); forprime(p=n^2+1, 2*n^2-1, if (issquare(p-1), nb++)); nb; \\ Michel Marcus, Feb 22 2023

Crossrefs

Cf. A000290, A002496, A060715.

Sequence in context: A339932 A254690 A156642 * A155124 A138033 A283876

Adjacent sequences: A360818 A360819 A360820 * A360822 A360823 A360824

Keyword

nonn

Author

Michel Lagneau, Feb 21 2023

Status

approved