A393333 Nonnegative integers k such that the Chebyshev distance between the vector of proportions of the digits 0-9 in 2^k and the uniform distribution (1/10, ..., 1/10) sets a new minimum.

0, 4, 7, 10, 14, 79, 87, 113, 119, 236, 246, 398, 646, 1022, 1245, 1537, 2440, 2467, 4063, 5003, 5175, 8137, 8695, 10694, 15793, 16549, 17783, 18725, 19010, 19431, 20211, 20900, 27197, 29458, 35503, 45368, 46269, 50150, 58656, 58723, 61743, 62107, 63258, 71498

Offset

1,2

Comments

The corresponding sequence for base 3 is finite: 0, 4, 5, 7, 13, 32, 33, 34, 66; this is because the ternary representation of 2^66 = 2^A117306(1) contains equally many 0's, 1's and 2's.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..66 (terms 1..62 from Pontus von Brömssen)

Example

   |      | frequencies f_i of 0-9 in 2^a(n) |  max |f_i/d-1/10|
 n | a(n) |   0  1  2  3  4  5  6  7  8  9   | (d = A034887(a(n)))
---+------+----------------------------------+---------------------
 1 |   0  |   0  1  0  0  0  0  0  0  0  0   |  9/10  = 0.9
 2 |   4  |   0  1  0  0  0  0  1  0  0  0   |  2/5   = 0.4
 3 |   7  |   0  1  1  0  0  0  0  0  1  0   |  7/30  = 0.233333...
 4 |  10  |   1  1  1  0  1  0  0  0  0  0   |  3/20  = 0.15
 5 |  14  |   0  1  0  1  1  0  1  0  1  0   |  1/10  = 0.1
 6 |  79  |   4  1  1  3  3  2  2  2  4  2   |  1/15  = 0.066666...
 7 |  87  |   3  2  4  3  4  4  2  2  1  2   | 17/270 = 0.062962...
 8 | 113  |   5  3  3  2  3  5  4  3  2  5   |  3/70  = 0.042857...
 9 | 119  |   3  4  3  4  4  3  4  3  3  5   |  7/180 = 0.038888...
10 | 236  |   8  5  7  6  9  6  6  9  9  7   | 11/360 = 0.030555...

Prog

(Python)
from itertools import groupby, count, islice
from gmpy2 import mpq, mpz, digits
def A393333_gen(): # generator of terms
    c, m, r = mpq(1), mpz(1), mpq(1, 10)
    for k in count(0):
        s = sorted(digits(m))
        d = len(s)
        f = max(max(abs(mpq(len(list(g)), d)-r) for a, g in groupby(s)), r*(len(set(s))<10))
        if f < c:
            c = f
            yield k
        m <<= 1
A393333_list = list(islice(A393333_gen(), 15)) # Chai Wah Wu, Feb 17 2026

Crossrefs

Cf. A027870, A034887, A065712, A065710, A065714-A065719, A065744, A117306, A393331 (Euclidean distance), A393332 (Manhattan distance).

Sequence in context: A393331 A393332 A243909 * A187334 A310698 A310699

Adjacent sequences: A393330 A393331 A393332 * A393334 A393335 A393336

Keyword

nonn,base

Author

Pontus von Brömssen, Feb 12 2026

Status

approved