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

2,1

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.

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.

Example

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

Crossrefs

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

Sequence in context: A090561 A273375 A016934 * A056229 A062271 A110258

Adjacent sequences: A260179 A260180 A260181 * A260183 A260184 A260185

Keyword

nonn,base

Author

Jon E. Schoenfield, Jul 17 2015

Status

approved