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A374267
Perfect squares whose pattern of identical digits is unique among the squares.
1
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.
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
KEYWORD
nonn,base
AUTHOR
Dmytro Inosov, Jul 02 2024
STATUS
approved