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%I #43 Jan 13 2021 20:52:27
%S -1,-1,225,-1,38025,314721,3111696,61058596,1026753849,31529329225,
%T 892067027049,-1,803752551280900,29501156485626049,
%U 1163446635475467225,-1,2200667320658951859841,104753558229986901966129,5272187100814113874556176,-1,15588378150732414428650569369
%N Smallest square which when written in base b contains each digit exactly once, or -1 if no such square exists.
%C 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"?
%C Does this sequence contain infinitely many positive terms? Equally, is A339693 infinite?
%C It is shown in A258103 that a(n) = -1 for n = 2,3,5,13,17,21 and infinitely many other values.
%H Chai Wah Wu, <a href="/A340501/b340501.txt">Table of n, a(n) for n = 2..29</a>
%e base a(base) digits
%e 4 225 [3, 2, 0, 1]
%e 6 38025 [4, 5, 2, 0, 1, 3]
%e 7 314721 [2, 4, 5, 0, 3, 6, 1]
%e 8 3111696 [1, 3, 6, 7, 5, 4, 2, 0]
%e 9 61058596 [1, 3, 6, 8, 0, 2, 5, 7, 4]
%e 10 1026753849 [1, 0, 2, 6, 7, 5, 3, 8, 4, 9]
%e 11 31529329225 [1, 2, 4, 0, 10, 5, 3, 6, 7, 8, 9]
%e 12 892067027049 [1, 2, 4, 10, 7, 11, 5, 3, 8, 6, 0, 9]
%e 14 803752551280900 [1, 0, 2, 6, 9, 11, 8, 12, 5, 7, 13, 3, 10, 4]
%o (Python)
%o from sympy import integer_nthroot
%o def digits(n, b):
%o out = []
%o while n >= b: n, r = divmod(n, b); out.append(r)
%o return [n] + out[::-1]
%o def a(n):
%o b, b2b = n, n**n
%o r, a = integer_nthroot(b**(b-1), 2); s = r**2
%o while s < b**(b-1): s += 2*r + 1; r += 1
%o while s < b2b:
%o if len(set(digits(s, b))) == n: return s
%o s += 2*r + 1; r += 1
%o return -1
%o print([a(n) for n in range(2, 13)]) # _Michael S. Branicky_, Jan 13 2021
%Y Inspired by A258103, A260182, A339693.
%Y Cf. A249034, A260191.
%K sign,base
%O 2,3
%A _N. J. A. Sloane_, Jan 13 2021
%E a(10)-a(22) from _Hugo Pfoertner_ and _Alois P. Heinz_, Jan 13 2021