A359440 A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.

0, 0, 0, 1, 2, 2, 1, 0, 0, 4, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0

Offset: 1

Author

Alexandre Herrera, Jan 01 2023

Keywords

• nonn

Comments

If the prime gaps above and below a prime p have the same length, p is called a balanced prime (see A006562). Likewise, if the prime gaps above and below the n-th prime gap have the same length, this gap might be called a balanced prime gap. These gaps correspond to nonzero terms a(n). Similarly, if a(n) >= 2, the n-th prime gap is the equivalent of a doubly balanced prime (A051795), and so on. - Peter Munn, Jan 08 2023

Examples

			For n = 1, prime(1) + prime(2) = 2 + 3 = 5; "prime(0)" does not exist, so a(1) = 0.
For n = 4:
  j = 0:  prime(4) + prime(5) =  7 + 11 = 18;
  j = 1:  prime(3) + prime(6) =  5 + 13 = 18;
  j = 2:  prime(2) + prime(7) =  3 + 17 = 20 != 18, so a(4) = 1.
For n = 5:
  j = 0:  prime(5) + prime(6) = 11 + 13 = 24;
  j = 1:  prime(4) + prime(7) =  7 + 17 = 24;
  j = 2:  prime(3) + prime(8) =  5 + 19 = 24;
  j = 3:  prime(2) + prime(9) =  3 + 23 = 26 != 24, so a(5) = 2.
		

Crossrefs

Cf. A000040, A006562, A051795, A055381, A081235.

Programs

Python

import sympy
offset = 1
N = 100
l = []
for n in range(offset,N+1):
    j = 0
    first_sum = sympy.prime(n-j)+sympy.prime(n+j+1)
    while (n-j) > 1:
        j += 1
        sum = sympy.prime(n-j)+sympy.prime(n+j+1)
        if sum != first_sum:
            break
    l.append(max(0,j-1))
print(l)

Formula

a(n) = min( {n-1} U {k : 0 <= k <= n-2 and prime(n-k-1) + prime(n+k+2) <> prime(n) + prime(n+1)} ). - Peter Munn, Jan 08 2023

Extensions

Introductory phrase added to name by Peter Munn, Jan 08 2023