A374176 -id:A374176 - cp's OEIS frontend

A373745 Maximum length of a run of alternating bits in the base-2 representation of prime(n).

2, 1, 3, 1, 3, 3, 2, 2, 3, 3, 1, 4, 4, 5, 3, 5, 3, 3, 2, 2, 3, 2, 4, 3, 2, 4, 2, 5, 3, 2, 1, 2, 3, 4, 6, 4, 3, 4, 4, 5, 3, 5, 3, 2, 4, 2, 4, 3, 2, 4, 4, 3, 2, 3, 2, 2, 3, 2, 6, 2, 3, 4, 2, 3, 2, 3, 4, 6, 5, 5, 3, 3, 3, 5, 3, 3, 4, 3, 3, 2, 4, 4, 5, 3, 3, 3, 2, 3, 3, 2, 4, 3, 2, 5

Offset: 1

James S. DeArmon, Jun 16 2024

			149 = prime(35) = 10010101_2 has two alternating bit runs of lengths 2 and 6: 10_010101, and thus a(35) = 6.

Cf. A000040, A374176.

b:= n-> `if`(n<2, [n$2], (f-> (t-> [t, max(t, f[2])])(
        `if`(n mod 4 in {0, 3}, 1, f[1]+1)))(b(iquo(n, 2)))):
a:= n-> b(ithprime(n))[2]:
seq(a(n), n=1..94);  # Alois P. Heinz, Jul 08 2024

from sympy import prime
def A373745(n):
    s = bin(prime(n))[2:]
    return next(i for i in range(len(s),0,-1) if ('01'*(i+1>>1))[:i] in s or ('10'*(i+1>>1))[:i] in s) # Chai Wah Wu, Jul 10 2024

a(n) = A374176(A000040(n)) + 1. - Alois P. Heinz, Jul 10 2024

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.

Original entry on oeis.org

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

Jean-Marc Rebert, Jul 07 2024

			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.

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

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)))

cp's OEIS Frontend

A373745 Maximum length of a run of alternating bits in the base-2 representation of prime(n).

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