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A260182
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Smallest square that is pandigital in base n.
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6
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4, 64, 225, 5329, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, 307197306432025, 803752551280900, 29501156485626049, 1163446635475467225, 830482914641378019961, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176
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OFFSET
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2,1
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COMMENTS
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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]
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.
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.
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LINKS
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EXAMPLE
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Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms begin as follows:
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n a(n) in base 10 a(n) in base n
== ========================= ======================
2 4 100_2
3 64 2101_3
4 225 3201_4
5 5329 132304_5
6 38025 452013_6
7 314721 2450361_7
8 3111696 13675420_8
9 61058596 136802574_9
10 1026753849 1026753849_10
11 31529329225 1240a536789_11
12 892067027049 124a7b538609_12
13 307197306432025 10254773ca86b9_13
14 803752551280900 10269b8c57d3a4_14
15 29501156485626049 102597bace836d4_15
16 1163446635475467225 1025648cfea37bd9_16
17 830482914641378019961 101246a89cgfb357ed_17
18 2200667320658951859841 10236b5f8eg4ad9ch7_18
19 104753558229986901966129 10234dhbg7ci8f6a9e5_19
20 5272187100814113874556176 1024e7cdi3hb695fja8g_20
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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