A340501 Smallest square which when written in base b contains each digit exactly once, or -1 if no such square exists.

-1, -1, 225, -1, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, -1, 803752551280900, 29501156485626049, 1163446635475467225, -1, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176, -1, 15588378150732414428650569369

Offset: 2

Author

N. J. A. Sloane, Jan 13 2021

Keywords

• sign
• base

Comments

Note that "pandigital" just means every digit appears at least once. The condition here is stronger. Maybe this should be called "Smallest strictly pandigital square in base b"?

Does this sequence contain infinitely many positive terms? Equally, is A339693 infinite?

It is shown in A258103 that a(n) = -1 for n = 2,3,5,13,17,21 and infinitely many other values.

Examples

			  base       a(base)  digits
   4             225 [3, 2, 0, 1]
   6           38025 [4, 5, 2, 0, 1, 3]
   7          314721 [2, 4, 5, 0, 3, 6, 1]
   8         3111696 [1, 3, 6, 7, 5, 4, 2, 0]
   9        61058596 [1, 3, 6, 8, 0, 2, 5, 7, 4]
  10      1026753849 [1, 0, 2, 6, 7, 5, 3, 8, 4, 9]
  11     31529329225 [1, 2, 4, 0, 10, 5, 3, 6, 7, 8, 9]
  12    892067027049 [1, 2, 4, 10, 7, 11, 5, 3, 8, 6, 0, 9]
  14 803752551280900 [1, 0, 2, 6, 9, 11, 8, 12, 5, 7, 13, 3, 10, 4]
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..29

Crossrefs

Inspired by A258103, A260182, A339693.

Cf. A249034, A260191.

Programs

Python

from sympy import integer_nthroot
def digits(n, b):
  out = []
  while n >= b: n, r = divmod(n, b); out.append(r)
  return [n] + out[::-1]
def a(n):
  b, b2b = n, n**n
  r, a = integer_nthroot(b**(b-1), 2); s = r**2
  while s < b**(b-1): s += 2*r + 1; r += 1
  while s < b2b:
    if len(set(digits(s, b))) == n: return s
    s += 2*r + 1; r += 1
  return -1
print([a(n) for n in range(2, 13)]) # Michael S. Branicky, Jan 13 2021

Extensions

a(10)-a(22) from Hugo Pfoertner and Alois P. Heinz, Jan 13 2021