A260182 Smallest square that is pandigital in base n.

4, 64, 225, 5329, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, 307197306432025, 803752551280900, 29501156485626049, 1163446635475467225, 830482914641378019961, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176

Offset: 2

Author

Jon E. Schoenfield, Jul 17 2015

Keywords

• nonn
• base

Comments

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.

Examples

			Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms begin as follows:
.
   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
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..28 (n = 2..22 from Jon E. Schoenfield)
• Rosetta Code, First perfect square in base N with N unique digits, lists a(n) for n = 2..39.

Crossrefs

Cf. A007953, A049363, A096008, A249034, A260117, A260191.