A258103 Number of pandigital squares (containing each digit exactly once) in base n.

0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562

Offset

2,6

Comments

For n = 18, the smallest and largest pandigital squares are 2200667320658951859841 and 39207739576969100808801. For n = 19, they are 104753558229986901966129 and 1972312183619434816475625. For n = 20, they are 5272187100814113874556176 and 104566626183621314286288961. - Chai Wah Wu, May 20 2015

When n is even, (n-1) is a factor of the pandigital squares. When n is odd, (n-1)/2 is a factor with the remaining factors being odd. Therefore, when n is odd and (n-1)/2 has an odd number of 2s as prime factors there are no pandigital squares in base n (e.g. 5, 13, 17 and 21). - Adam J.T. Partridge, May 21 2015

Links

Table of n, a(n) for n=2..19.

A. J. T. Partridge, Why there are no pandigital squares in base 13

Example

For n=4 there is one pandigital square, 3201_4 = 225 = 15^2.
For n=6 there is one pandigital square, 452013_6 = 38025 = 195^2.
For n=10 there are 87 pandigital squares (A036745).
There are no pandigital squares in bases 2, 3, 5 or 13.
Hexadecimal has 3402 pandigital squares, the largest is FED5B39A42706C81.

Prog

(Python)
from gmpy2 import isqrt, mpz, digits
def A258103(n): # requires 2 <= n <= 62
....c, sm, sq = 0, mpz(''.join([digits(i, n) for i in range(n-1, -1, -1)]), n), mpz(''.join(['1', '0']+[digits(i, n) for i in range(2, n)]), n)
....m = isqrt(sq)
....sq = m*m
....m = 2*m+1
....while sq <= sm:
........if len(set(digits(sq, n))) == n:
............c += 1
........sq += m
........m += 2
....return c # Chai Wah Wu, May 20 2015

Crossrefs

Cf. A036745, A054038, A071519.

Sequence in context: A004206 A302397 A151372 * A300373 A222112 A032832

Adjacent sequences: A258100 A258101 A258102 * A258104 A258105 A258106

Keyword

base,nonn,more

Author

Adam J.T. Partridge, May 20 2015

Extensions

a(17)-a(19) from Giovanni Resta, May 20 2015

Status

approved