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%I #40 Dec 08 2019 04:46:25
%S 4,64,225,5329,38025,314721,3111696,61058596,1026753849,31529329225,
%T 892067027049,307197306432025,803752551280900,29501156485626049,
%U 1163446635475467225,830482914641378019961,2200667320658951859841,104753558229986901966129,5272187100814113874556176
%N Smallest square that is pandigital in base n.
%C Compare this sequence with A260117, Smallest triangular number that is pandigital in base n. Presumably, lim_{n->infinity} A260117(n)/A049363(n) = 1, but the same cannot be true for this sequence: the sum of the base-n digits of a number that is pandigital in base n must be 0+1+2+...+n-1 = binomial(n,2), but there are certain values of n for which no n-digit square can have a digit sum of binomial(n,2); for such values of n, a(n) must have more than n digits in base n. [E.g., the base-13 expansion of every square has a digit sum s == {0,1,4,9} (mod 12) (cf. A096008), but a square that is pandigital in base 13 and has exactly 13 digits would have a digit sum s = 78 == 6 (mod 12), so no such number exists; a 14-digit base-13 pandigital square would have each of the digits 0..12 exactly once except for one duplicated digit, which would have to be 3, 6, 7, or 10 (to yield a digit sum of 81, 84, 85, or 88, whose residues modulo 12 are 9, 0, 1, and 4, respectively). - _Jon E. Schoenfield_, Mar 23 2019]
%C The values of n for which there exists no pandigital square that is exactly n digits long (in base n) begin with 2, 3, 5, 13, 17, 21, ...; presumably, for all such values of n, a(n) is exactly n+1 base-n digits long.
%C In base 2, there are no 2-digit squares at all, so a(2) must have more than 2 binary digits. For n = 3, 5, 13, 17, 21, ..., there exists no square, regardless of its number of digits, whose base-n digit sum equals binomial(n,2); see A260191.
%H Chai Wah Wu, <a href="/A260182/b260182.txt">Table of n, a(n) for n = 2..28</a> (n = 2..22 from Jon E. Schoenfield)
%H Rosetta Code, <a href="http://rosettacode.org/wiki/First_perfect_square_in_base_N_with_N_unique_digits">First perfect square in base N with N unique digits</a>, lists a(n) for n = 2..39.
%e Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms begin as follows:
%e .
%e n a(n) in base 10 a(n) in base n
%e == ========================= ======================
%e 2 4 100_2
%e 3 64 2101_3
%e 4 225 3201_4
%e 5 5329 132304_5
%e 6 38025 452013_6
%e 7 314721 2450361_7
%e 8 3111696 13675420_8
%e 9 61058596 136802574_9
%e 10 1026753849 1026753849_10
%e 11 31529329225 1240a536789_11
%e 12 892067027049 124a7b538609_12
%e 13 307197306432025 10254773ca86b9_13
%e 14 803752551280900 10269b8c57d3a4_14
%e 15 29501156485626049 102597bace836d4_15
%e 16 1163446635475467225 1025648cfea37bd9_16
%e 17 830482914641378019961 101246a89cgfb357ed_17
%e 18 2200667320658951859841 10236b5f8eg4ad9ch7_18
%e 19 104753558229986901966129 10234dhbg7ci8f6a9e5_19
%e 20 5272187100814113874556176 1024e7cdi3hb695fja8g_20
%Y Cf. A007953, A049363, A096008, A249034, A260117, A260191.
%K nonn,base
%O 2,1
%A _Jon E. Schoenfield_, Jul 17 2015
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