A385924 Integers k such that 2^k-1 has 0 as its second-most significant digit in base k.

10, 19, 39, 181, 493, 941, 2454, 16111, 30730, 416488, 1433896, 1663074

Offset: 1

Author

A. Lamek, Jul 12 2025

Keywords

• nonn
• base
• more

Comments

Inspired by the Wieferich primes which ask for two least significant 0 digits of 2^(p-1)-1 in base p.

Examples

			19 is a term since 2^19-1 = 524287 = 40861_19 has 0 as its second-most significant base 19 digit.
		

Crossrefs

Cf. A000225, A001220 (Wieferich primes).

Programs

Mathematica

isok[k_] := Module[{d = IntegerDigits[2^k - 1, k]}, Length[d] >= 2 && d[[2]] == 0]

PARI

isok(k) = digits(2^k-1, k)[2] == 0; \\ Michel Marcus, Jul 17 2025

PARI

isok(k) = if (k>1, my(X=2^k-1); !((X\k^(logint(X, k)-1)) % k)); \\ Michel Marcus, Jul 17 2025

Python

def is_a385924(k):
    x = pow(2, k) - 1
    r = None
    while x >= k:
        r = x % k
        x //= k
    return r == 0

Python

def is_a385924(k):
    x = pow(2, k) - 1
    approx = k // k.bit_length()
    power = pow(k, approx)
    while power * k <= x:
        power *= k
        approx += 1
    return (x // pow(k, approx - 1)) % k == 0

Extensions

a(11)-a(12) from Michael S. Branicky, Jul 20 2025