A374402 Least number that is the lesser of two consecutive primes p and q whose binary expansions have the same length and agree at exactly n digit positions, or -1 if no such prime pair exists.

2, 5, 23, 17, 41, 67, 137, 269, 521, 1049, 2081, 4111, 8233, 16417, 32771, 65537, 131113, 262147, 524309, 1048609, 2097257, 4194389, 8388617, 16777289, 33554501, 67109123, 134217929, 268435459, 536871017, 1073741827, 2147484041, 4294967497, 8589934627, 17179869731

Offset: 1

Author

Jean-Marc Rebert, Jul 07 2024

Keywords

• nonn
• base

Examples

			a(1) = 2 because 2 = 10_2 and 3 = 11_2 are two consecutive primes that, when written in base 2, both have 2 digits and agree at exactly 1 digit position (each has a 1 in its first digit position), and no earlier pair of consecutive primes has this property.
a(3) = 23 = 10111_2; the next prime is
       29 = 11101_2  (same number of binary digits),
            ^ ^ ^    and the digits agree at 3 digit positions,
  and no earlier pair of consecutive primes has this property.
		

Crossrefs

Cf. A000040, A006879, A205510, A319840, A006879, A319840, A339080, A374176.

Programs

PARI

card(p)=my(u=binary(p),v=binary(nextprime(p+1))); if(#u!=#v,return(0)); sum(i=1,#u,u[i]==v[i])
a(n)=forprime(p=2^n,oo,if(card(p)==n,return(p)))