A060757 -id:A060757 - cp's OEIS frontend

A382481 a(n) is the number of primes less than 4^(n^2).

0, 2, 54, 23000, 203280221, 33483379603407, 96601075195075186855

Offset: 0

Stefano Spezia, Mar 28 2025

Thue's proof, that there exist infinitely many primes, shows also that n+1 is a lower bound for a(n) (see Ribenboim, 2004).

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 9-10.

a(n) = pi(4^(n^2)).
a(n) = A007053(2*n^2). - _Amiram Eldar_, Mar 29 2025

Cf. A000290, A000302, A000720, A007053, A060757.

Mathematica
```
a[n_]:=PrimePi[4^(n^2)]; Array[a,6,0]
```

from sympy import primepi
def A382481(n): return primepi(1<<(n**2<<1)) # Chai Wah Wu, Mar 29 2025

A383255 Number of n X n {0,1,2,3} matrices having no 1's to the right of any 0's and no 3's above any 2's.

Original entry on oeis.org

1, 4, 194, 107080, 672498596, 48104236145168, 39202958861329453384, 364022757339778569993689888, 38513979937284562006371342202842000, 46429021191757554279412904483559912259714112, 637737721080296383894709847744103523361428384973270816

Offset: 0

John Tyler Rascoe, Apr 20 2025

nonn

These are matrices with no [0,1] or [3] submatrices.

[2]

			The 3 X 3 matrices below are counted under a(3) = 107080:
 [0,0,0] [1,0,2] [2,3,2]
 [0,0,0] [1,0,3] [3,3,3]
 [0,0,0],[0,2,3],[3,3,3].

John Tyler Rascoe, Python program.

Cf. A002416, A006506, A014235, A060757, A181213, A213977, A381857.

Python
```
# see links
```

a(n) <= A060757(n).

a(5)-a(10) from Bert Dobbelaere, Apr 23 2025

cp's OEIS Frontend

A382481 a(n) is the number of primes less than 4^(n^2).

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