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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.
0
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
1,1
EXAMPLE
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.
PROG
(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)))
KEYWORD
nonn,base
AUTHOR
Jean-Marc Rebert, Jul 07 2024
STATUS
approved