A347986 Length of identical sequence of prime / nonprime numbers left and right of the integer n (excluded).

0, 0, 0, 1, 2, 2, 1, 0, 5, 0, 1, 8, 1, 0, 9, 0, 1, 6, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 18, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 12, 1, 0, 3, 0, 3, 0, 1, 6, 1, 0, 7, 0, 1, 4, 1, 0, 1, 10, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 4, 1, 0, 3, 0, 9, 0, 3, 0, 1, 6, 1, 0, 5, 0, 1, 2, 7, 2, 1, 0, 3, 0, 3

Offset: 1

Author

Lothar Esser, Sep 22 2021

Keywords

• nonn

Examples

			For n=2 (first useful term) the result is 0 because 2 is preceded by 1 which is by definition nonprime and succeeded by 3 which is prime meaning symmetry is broken right away. A better example may be 5 with a value of 2. The two numbers preceding 5 are 3, 4: prime, nonprime and the succeeding values are 6 and 7 being nonprime and prime. In other words, starting from 5 as center, the first positions are 4 (left) and 6 (right), both nonprimes. The next positions are 3 and 7, both primes. The sequence is now 2 long. It breaks after that because 2 is prime but 8 is nonprime. So we note 2 or 5. Very interesting is 30 which has a sequence of 18 on each side that follow the same pattern.
From _Jon E. Schoenfield_, Sep 22 2021: (Start)
As shown in the illustration below, where P and N denote prime and nonprime, respectively, the distribution of primes and nonprimes around n=21 is symmetrical in the interval [18, 24] = [21-3, 21+3], but not in the interval [17, 25] = [21-4, 21+4] (since 17 is prime but 25 is composite), so a(21) = 3:
.
              |<------- 3 -------->|<------- 3 -------->|
      17     18     19     20     21     22     23     24     25
  -----+------+------+------+------+------+------+------+------+-----
       P      N      P      N      N      N      P      N      N
              |      |      |             |      |      |
              |      |      +-------------+      |      |
              |      +---------------------------+      |
              +-----------------------------------------+
(End)
		

Crossrefs

Cf. A343730, A346399 (number of primes in the interval n +- a(n)).

Programs

Mathematica

Table[s={n-1,n+1};k=0;While[SameQ@@PrimeQ@s,k++;s=s+{-1,+1}];k,{n,2,85}] (* Giorgos Kalogeropoulos, Sep 23 2021 *)

PARI

f(n) = {my (nb = 0, fL, fR); fL = n\2; if (n%2, fR = fL+1, fL--; fR = fL+2); for (i=0, oo, if (isprime(fL-i) != isprime(fR+i), break, nb++);); if (n%2, 2*nb, 2*nb+1);} \\ A343730
a(n) = (f(2*n) - 1)/2; \\ Michel Marcus, Sep 24 2021

Python

from sympy import *
seq_pole = []
seq_pole.append(-1) #0
seq_pole.append(-1) #1
for i in range(1, 1000):
    d = 1
    #  Check how far the left is identical to the
    # (mirrored) right
    while isprime(i-d) == isprime(i+d):
        d = d + 1
    dmax = d - 1
    seq_pole.append(dmax)
    # i is the center (index) and dmax is the max extent
    # on each side that is the same (or mirrored at i if you will)
    print("{}".format(dmax))

Formula

From Jon E. Schoenfield, Sep 22 2021: (Start)

a(n) = Max_{k : A010051(n-j) = A010051(n+j) for 0 <= j <= k}.

a(n) = (A343730(2*n) - 1)/2. (End)

Extensions

More terms from Michel Marcus, Sep 24 2021