A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

2, 2039, 6143, 522239, 33546239, 260046847, 16911433727, 32212254719, 2196875771903, 140735340871679, 2251799813685119, 9005000231485439, 576460752169205759, 36893488147410714623, 147573811852188057599, 9444732965739282038783, 154742504910672534362390399

Offset: 1

Author

Ya-Ping Lu, Dec 23 2023

Keywords

• base
• nonn

Comments

It seems that most of the terms end with '9', followed by those ending with '3', '7', and '1'.

Examples

			2 is a term because 2 is a prime with one '0' in binary form ('10') and '01' is not a prime. 2039 is a term because 2039 is a prime with one '0' in binary form ('11111110111') and changing the position of the '0', for example, '11111111011' = 2043 and '01111111111' = 1023, always results in a composite.
		

Crossrefs

Subsequence of A095078.

Cf. A039986, A081118, A095058, A138290, A208083.

Programs

Python

from sympy import isprime
for n in range(1,100):
    s = n*'1'; c = 0
    for j in range(n+1):
        num = int(s[:j]+'0'+s[j:], 2)
        if isprime(num):
            c += 1
            if c == 1: r = num
            if c == 2: break
    if c == 1: print(r, end = ', ')