A374267 Perfect squares whose pattern of identical digits is unique among the squares.

1444, 7744, 14884, 19881, 29929, 37636, 40401, 44944, 46656, 55696, 66564, 69696, 116964, 133225, 136161, 144400, 166464, 190969, 202500, 219961, 224676, 225625, 261121, 276676, 277729, 300304, 339889, 407044, 438244, 473344, 511225, 525625, 544644, 553536, 555025, 556516, 585225

Offset

1,1

Comments

The digit pattern for any natural number n is uniquely defined by the canonical form A358497(n), which enumerates digits in order of their first occurrence in n, from left to right.

Each perfect square in this sequence has a unique digit pattern in the sense that no other square has the same pattern.

A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.If a cryptarithmetic problem calls for a perfect square, then the squares in this sequence are unique solutions, so we call them cryptarithmically unique.

Links

Table of n, a(n) for n=1..37.

Wikipedia, Verbal arithmetic.

Formula

a(n) = A374268(n)^2.

Example

The first cryptarithmically unique square is 38^2=1444. This means that no other square has the same digit pattern "ABBB".
Counterexample: 144=12^2 is not in this sequence because 400=20^2 is also a perfect square with the same digit pattern "ABB". Equivalently, A358497(144)=A358497(400)=122.
The alphametic puzzle SEA^2 = BIKINI has a solution 437^2 = 190969 (K=0, B=1, E=3, S=4, N=6, A=7, I=9). This solution is unique because 190969 is a term in this sequence.

Mathematica

NumOfDigits = 4; (* Maximal integer length to be searched for *)
A358497[k_] := With[{pI = Values@PositionIndex@IntegerDigits@k}, MapIndexed[#1 -> Mod[#2[[1]], 10] &, pI, {2}] // Flatten // SparseArray // FromDigits];
Extract[Extract[Select[Tally[Table[{#, A358497[#]} &[i^2], {i, 1, 10^NumOfDigits - 1}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}], {All, 1}]

Crossrefs

Subsequence of A000290 (squares).

Cf. A374268 (bases of cryptarithmically unique squares).

Cf. A374238 (cryptarithmically unique primes).

Sequence in context: A348740 A368191 A298222 * A180465 A328886 A031754

Adjacent sequences: A374264 A374265 A374266 * A374268 A374269 A374270

Keyword

nonn,base

Author

Dmytro Inosov, Jul 02 2024

Status

approved