A363873 Least k such that 2^k begins with n but is not exactly n.

4, 8, 5, 12, 9, 6, 46, 13, 53, 10, 50, 7, 17, 47, 77, 14, 34, 54, 84, 11, 31, 51, 61, 81, 8, 18, 38, 48, 68, 78, 98, 15, 25, 35, 45, 55, 75, 85, 95, 12, 22, 32, 42, 145, 52, 62, 72, 82, 92, 102, 9, 19, 29, 39, 142, 49, 59, 162, 69, 79, 89, 192, 99, 109, 16, 119, 26, 36, 139, 46

Offset: 1

Author

Robert G. Wilson v, Jul 03 2023

Keywords

• nonn
• base

Comments

This is not an injective function. a(2) = a(25) = 8.

a(n) > 3.

Examples

			a(1) = 4 since 2^4 = 16 starts with 1 and is not 1 itself (the way 2^0 = 1 would be);
a(2) = 8 (not 1: 2^1 = 2) since 2^8 = 256;
a(3) = 5 since 2^5 = 32;
a(4) = 12 (not 2: 2^2 = 4) since 2^12 = 4096;
a(5) = 9 since 2^9 = 512; etc.
		

Crossrefs

Cf. A000079, A018856.

Programs

Mathematica

a[n_] := Block[{j = IntegerLength@ n, k = 1}, While[ IntegerLength[2^k] < j || Quotient[2^k, 10^(IntegerLength[2^k] - j)] != n || n == 2^k, k++]; k]; Array[ a, 70]

Python

def A363873(n):
    m, s = 1<<(k:=n.bit_length()-1), str(n)
    while m<=n or not str(m).startswith(s):
        k += 1
        m <<= 1
    return k # Chai Wah Wu, Aug 06 2023