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A258103 Number of pandigital squares (containing each digit exactly once) in base n. 1
0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562 (list; graph; refs; listen; history; text; internal format)
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: A055348 A004206 A151372 * A222112 A032832 A041021

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

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Last modified August 19 08:57 EDT 2017. Contains 290794 sequences.