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A343730
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a(n) is the length of the longest run of consecutive integers centered at n/2 with primality symmetric about n/2, or 0 if no such run exists.
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2
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2, 1, 0, 1, 4, 1, 0, 3, 0, 5, 0, 5, 0, 3, 0, 1, 2, 11, 2, 1, 0, 3, 0, 17, 0, 3, 0, 1, 2, 19, 2, 1, 0, 3, 0, 13, 0, 3, 0, 1, 2, 7, 2, 1, 0, 7, 0, 1, 2, 3, 4, 9, 4, 3, 2, 1, 0, 3, 0, 37, 0, 3, 0, 1, 2, 3, 4, 9, 4, 3, 2, 1, 0, 7, 0, 1, 2, 7, 2, 1, 0, 3, 0, 25, 0
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OFFSET
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1,1
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COMMENTS
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Consider the infinite string S of binary digits whose n-th digit is 1 iff n is prime, i.e.,
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S = 01101010001010001010001000001010000010001010001000...
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The 37-digit substring centered at the 30th digit, i.e.,
0100010100010000010100000100010100010
(corresponding to n = 12, ..., 48) is palindromic; equivalently, 30-k and 30+k are both prime or both nonprime for each k in the interval [0, 18]. However, the 39-digit substring with the same center is not palindromic; it starts with a 1 and ends with a 0, since 30 - 19 = 11 is prime while 30 + 19 = 49 is not.
a(n) is the length of the longest palindromic substring in S that is centered at digit position n/2.
a(n) is odd when n is even and vice versa.
For even n, the consecutive integer "run" of length 1 consisting only of the integer n/2 always has primality symmetric about its center, so a(n) >= 1 for n even.
For odd n, the run of length 2 consisting of the integers j1 = (n-1)/2 and j2 = (n+1)/2 has primality symmetric about its center iff j1 and j2 are both prime (which occurs only at n = 5) or both nonprime (which occurs for all values of n in A166685 except for A166685(2)=5). For all odd n not in A166685, j1 and j2 are, in some order, a prime and a nonprime, so no run with primality symmetric about the center exists, so a(n) = 0.
The even values > 4 tend to make their first appearance in the sequence much later than nearby odd values. E.g., the first 5, 7, 9, 11, and 13 appear at n = 10, 42, 52, 18, and 36, but the first 6, 8, 10, 12, and 14 do not appear until n = 185, 235, 237, 239, and 1061, respectively.
For n >= 7, a(n) <= n - 5 since the only occurrence of 11 in S is in digits 2 and 3. - Michael S. Branicky, Sep 23 2021
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LINKS
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EXAMPLE
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For n=1, the shortest run of consecutive integers centered at 1/2 is {0, 1}; both are nonprime, so its primality is symmetric about its center. The next longer run of consecutive integers centered at 1/2 is {-1, 0, 1, 2}; 2 is prime, but -1 is not, so this run's primality is not symmetric about its center, and the same will be true of any longer run centered at 1/2 (e.g., {-2, -1, 0, 1, 2, 3}). So the longest run with primality symmetric about 1/2 is {0, 1}, whose length is 2, so a(1)=2.
For n=2, the "run" of length 1 centered at 2/2 = 1 is simply {1} (and, like every run of length 1, has primality symmetric about its center). The run of length 3 centered at 1 is {0, 1, 2}; 2 is prime, but 0 is not, so a(2)=1.
For n=3, the shortest run of consecutive integers centered at 3/2 is {1, 2}, whose primality is not symmetric about the center (2 is prime while 1 is not), and the same will be true of any longer run centered at 3/2, so no run with primality symmetric about 3/2 exists, so a(3)=0.
For n=5, the run {2, 3} has symmetric primality (both 2 and 3 are prime), and so does {1, 2, 3, 4} (both 1 and 4 are nonprime), but {0, 1, 2, 3, 4, 5} does not (5 is prime, 0 is not), so a(5)=4.
For n=18, the run of length 11 centered at 18/2 = 9 is
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
with primality 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0,
which is symmetric, but the run of length 13 is not (since 3 is prime while 15 is not), so a(18)=11.
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MATHEMATICA
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Table[s=If[OddQ@n, {Floor[n/2], Ceiling[n/2]}, {n/2-1, n/2+1}]; k=0; While[SameQ@@PrimeQ@s, k++; s=s+{-1, +1}]; If[OddQ@n, 2k, 2k+1], {n, 85}] (* Giorgos Kalogeropoulos, Sep 23 2021 *)
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PROG
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(PARI) a(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); } \\ Michel Marcus, Sep 23 2021
(Python)
from sympy import isprime
def ispal(s):
return s == s[::-1]
def primestr(a, b):
return "".join('1' if isprime(k) else '0' for k in range(a, b+1))
def a(n):
fl, cg = n//2, (n+1)//2
start, end, r = fl, cg, n%2-1
while ispal(primestr(start, end)):
start, end, r = start-1, end+1, r+2
return r
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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