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

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

KEYWORD

nonn,base

AUTHOR

Dmytro Inosov, Jul 02 2024

STATUS

approved