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

Author

James S. DeArmon, Jun 16 2024

Keywords

• nonn
• base
• easy

Examples

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

Crossrefs

Cf. A000040, A374176.

Programs

Maple

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

Python

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

Formula

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