A364714 Least positive integer whose average digit in base b equals (b-1)/2 (the expected value for random digits) for 2 <= b <= n.

2, 38, 141, 3468, 36990, 36990, 789730327537467540, 789730327537467540, 789731071815355740, 789731549802436500

Offset

2,1

Comments

a(n) has an even number of digits in all even bases b <= n.

a(8) <= 795482814912042180, a(9) and a(10) <= 836119295625913740. - Giorgos Kalogeropoulos, Aug 09 2023

a(8) and a(9) <= 789730327537467540, a(10) <= 789731071815355740, a(11) <= 789731549802436500. - Jason Yuen, May 17 2024

a(8) > A144812(10000) = 16960567248690 (last term in b-file for A144812). - Pontus von Brömssen, May 19 2024

Links

Table of n, a(n) for n=2..11.

2025 Canadian Computing Olympiad Committee, 2025 Canadian Computing Olympiad, Day 1, Problem 3: Balanced Integer, University of Waterloo.

Example

For n = 4, 141 is 10001101 in binary with average digit 1/2, 12020 in base 3 with average digit 2/2 = 1, and 2031 in base 4 with average digit 3/2. No smaller number has this property, so a(4) = 141.

Prog

(PARI) isokb(k, b) = my(d=digits(k, b)); vecsum(d)/#d == (b-1)/2;
isok(k, n) = for (b=2, n, if (!isokb(k, b), return(0)); ); 1;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Aug 05 2023

Crossrefs

a(2)-a(7) are the first terms of A031443, A144798, A144799, A144800, A144801, and A144812, respectively.

Sequence in context: A075459 A050248 A337773 * A105645 A098456 A214909

Adjacent sequences: A364711 A364712 A364713 * A364715 A364716 A364717

Keyword

nonn,base,more,hard

Author

Pontus von Brömssen, Aug 04 2023

Extensions

a(8)-a(11) from Jason Yuen, Oct 18 2025

Status

approved