A389533 a(n) is the largest number m in base n such that the digits of the pair (m, 2*m) use every digit in base n exactly once; a(n) = 0 if no such number exists.

0, 0, 6, 14, 0, 223, 1953, 6036, 48651, 0, 1465849, 4608378, 0, 0, 2127254040, 6730515944, 0, 313220703669, 5090280057661, 16241087607090, 290775726273319, 930162093280607, 18188380249950025, 58310817710004452, 1236458658912184475, 3971607849480097073, 90771216766121660049

Offset

2,3

Comments

Pairs using leading 0s, for example (03,12) in base 4, are considered invalid.

Links

Andrew George Dent Clarke, Table of n, a(n) for n = 2..39

Michael S. Branicky, Python program

Andrew George Dent Clarke, Python Program

Andrew George Dent Clarke, Illustrative Table

Example

(a(7),2*a(7)) is written (436,1205) in base 7.

Prog

(PARI) a(n)={my(lb=if(n%2, n^(n\2)\2, n^(n/2-1)), ub=if(n%2, n^(n\2), n^(n/2)\2)); forstep(k=ub, lb, -1, my(v=concat(digits(k, n), digits(2*k, n))); if(#v==n && #Set(v)==n, return(k))); 0} \\ Andrew Howroyd, Oct 08 2025
(Python) # see linked programs to generate larger terms
from sympy.ntheory import digits
from itertools import permutations
def to_digits(k, base): return tuple(digits(k, base)[1:])
def from_digits(t, base): return sum(ti*base**i for i, ti in enumerate(t[::-1]))
def ok(t1, t2, base): return base == len(t1+t2) == len(set(t1+t2))
def a(n):
    first = list(range(n//2, n)) if n&1 else list(range(n//2))
    return next((k for f in first[::-1] for p in permutations(sorted(set(range(n))-{f})[::-1], n//2-1) if ok((f, )+p, to_digits(2*(k:=from_digits((f, )+p, n)), n), n)), 0)
print([a(n) for n in range(2, 14)]) # Michael S. Branicky, Oct 10 2025

Crossrefs

Sequence in context: A248263 A380093 A201449 * A205300 A377291 A065938

Adjacent sequences: A389530 A389531 A389532 * A389534 A389535 A389536

Keyword

base,nonn

Author

Andrew George Dent Clarke, Oct 07 2025

Status

approved