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

2,3

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.

Links

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

Example

  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]

Prog

(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

Crossrefs

Inspired by A258103, A260182, A339693.

Cf. A249034, A260191.

Sequence in context: A204977 A204704 A206649 * A370611 A265432 A247885

Adjacent sequences: A340498 A340499 A340500 * A340502 A340503 A340504

Keyword

sign,base

Author

N. J. A. Sloane, Jan 13 2021

Extensions

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

Status

approved