A374167 a(n) is the largest k such that the binary representation of prime(n), when interpreted as a base-b number, yields a prime for every b in 2..k.

3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 4, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 5, 2, 2, 2, 5, 2, 5, 5, 3, 2, 2, 3, 3, 3, 4, 5, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2

Offset

1,1

Comments

a(n) = 2 iff prime(n) is not in A235266.

Conjecture: every integer > 1 eventually appears.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = max{ k : Sum_{j=0..D} b^j * d_j is a prime for b = 2..k } where the digits of the binary representation of n are d_D, d_{D-1}, ..., d_1, d_0.

Example

a(1) = 3 because prime(1) = 2 = 10_2, and both 10_2 = 2 and 10_3 = 3 are primes, but 10_4 = 4 = 2*2.
a(36) = 5 because prime(36) = 151 = 10010111_2, and
  10010111_2 = 151 is a prime
  10010111_3 = 2281 is a prime,
  10010111_4 = 16661 is a prime, and
  10010111_5 = 78781 is a prime, but
  10010111_6 = 281275 = 5^2 * 11251.

Prog

(PARI) a(n) = my(v=binary(prime(n)), b=2); while (isprime(fromdigits(v, b)), b++); b-1; \\ Michel Marcus, Jul 02 2024

Crossrefs

Cf. A235266, A374264.

Sequence in context: A069901 A115039 A032536 * A115061 A217834 A227877

Adjacent sequences: A374164 A374165 A374166 * A374168 A374169 A374170

Keyword

nonn,base

Author

Jon E. Schoenfield, Jun 29 2024

Status

approved