A287298 a(n) is the largest square with distinct digits in base n.

1, 1, 225, 576, 38025, 751689, 10323369, 355624164, 9814072356, 279740499025, 8706730814089, 23132511879129, 11027486960232964, 435408094460869201, 18362780530794065025, 48470866291337805316, 39207739576969100808801, 1972312183619434816475625, 104566626183621314286288961

Offset: 2

Author

John L. Drost, May 22 2017

Keywords

• nonn
• base

Comments

a(n) does not always have n digits in base n. If n is 5 mod 8 then a number which contains all the digits in base n is congruent to (n-1)n/2 mod (n-1). It will be then divisible by a single power of 2 and not a square.

a(22) = 340653564758245010607213613056. - Chai Wah Wu, May 24 2017

Examples

			a(4)=225 which is 3201 in base 4. Higher squares have at least 5 digits in base 4.
		

Programs

Python

from gmpy2 import isqrt, mpz, digits
def A287298(n): # assumes n <= 62
    m = isqrt(mpz(''.join(digits(i,n) for i in range(n-1,-1,-1)),n))
    m2 = m**2
    d = digits(m2,n)
    while len(set(d)) < len(d):
        m -= 1
        m2 -= 2*m+1
        d = digits(m2,n)
    return int(m2) # Chai Wah Wu, May 24 2017

Extensions

Added a(16)-a(20) and corrected a(12) by Chai Wah Wu, May 24 2017