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

1,1

Comments

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

a(n) > 3.

Links

Table of n, a(n) for n=1..70.

Example

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.

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]

Prog

(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

Crossrefs

Cf. A000079, A018856.

Sequence in context: A124193 A276577 A011366 * A372355 A005133 A198241

Adjacent sequences: A363870 A363871 A363872 * A363874 A363875 A363876

Keyword

nonn,base

Author

Robert G. Wilson v, Jul 03 2023

Status

approved