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A340501 Smallest square which when written in base b contains each digit exactly once, or -1 if no such square exists. 2
-1, -1, 225, -1, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, -1, 803752551280900, 29501156485626049, 1163446635475467225, -1, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176, -1, 15588378150732414428650569369 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A204977 A204704 A206649 * A370611 A265432 A247885
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

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)