A365097 Smallest k > 1 such that the total number of digits "1" required to write the numbers 1..k in base n is equal to k.

2, 4, 25, 181, 421, 3930, 8177, 102772, 199981, 3179142, 5971945, 143610511, 210826981, 4754446846, 8589934561, 222195898593, 396718580701, 13494919482970, 20479999999961, 764527028941797, 1168636602822613, 41826814261329722, 73040694872113105, 2855533828630999398

Offset

2,1

Comments

a(10) = A014778(3), being the smallest term > 1 there.

An upper bound is a(n) <= A226238(n) = u, since the digits of u show there are u 1's in numbers 1..u (in base n). - Kevin Ryde, Sep 28 2023

Links

Jon E. Schoenfield, Table of n, a(n) for n = 2..200

Formula

For even n > 2, a(n) = 2*n^(n/2) - 2*n + 1. - Jon E. Schoenfield, Sep 30 2023

Example

For n=2, the first k=2 positive integers are 1 = 1_2 and 2 = 10_2, which have a total of two 1's, so a(2) = 2.
For n=3, the first k=4 positive integers, which are 1_3, 2_3, 10_3, and 11_3, have a total of four 1's, which is equal to k, so a(3) = 4.
For n=4, a total of 25 1's occur in the first k=25 positive integers (they occur in 1_4, 10_4, 11_4, 12_4, 13_4, 21_4, 31_4, 100_4, 101_4, 102_4, 103_4, 110_4, 111_4, 112_4, 113_4, 120_4, and 121_4 = 25), so a(4) = 25.

Mathematica

a[n_] := Module[{k = 1, sum = 1}, While[sum == 1 || sum != k, k++; sum += Count[IntegerDigits[k, n], 1]]; k]; Array[a, 6, 2] (* Amiram Eldar, Aug 29 2023 *)

Prog

(Python)
from itertools import count
from sympy.ntheory.factor_ import digits
def A365097(n):
    c, a, q, m = 1, 1, 0, 1
    for k in count(2):
        m += 1
        if m == n:
            m = 0
            q += 1
            a = digits(q, n).count(1)
        elif m==1:
            a += 1
        elif m==2:
            a -= 1
        c += a
        if c == k:
            return k # Chai Wah Wu, Sep 28 2023

Crossrefs

Cf. A014778, A094798, A226238.

Sequence in context: A103099 A342665 A266495 * A119029 A291144 A162120

Adjacent sequences: A365094 A365095 A365096 * A365098 A365099 A365100

Keyword

nonn,base

Author

Andrew Pope, Aug 21 2023

Extensions

a(11)-a(15) from Amiram Eldar, Aug 29 2023

a(16)-a(19) from Chai Wah Wu, Sep 29 2023

a(20)-a(25) from Jon E. Schoenfield, Sep 30 2023

Status

approved