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Numbers whose squares have a unique pattern of identical digits among the squares.

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`%I #37 Aug 03 2024 19:19:53
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`%S 38,88,122,141,173,194,201,212,216,236,258,264,342,365,369,380,408,
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`%T 437,450,469,474,475,511,526,527,548,583,638,662,688,715,725,738,744,
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`%U 745,746,765,796,804,813,816,836,880,893,898,908,970,995,1020
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`%N Numbers whose squares have a unique pattern of identical digits among the squares.
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`%C 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.
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`%C The square of each term in this sequence has a unique digit pattern in the sense that no other square has the same pattern.
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`%C A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.
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`%C If a cryptarithmetic problem calls for a perfect square, then the squares of numbers in this sequence are unique solutions, so we call them cryptarithmically unique.
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`%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Verbal_arithmetic">Verbal arithmetic</a>.
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`%F a(n) = sqrt(A374267(n)).
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`%e The first term of this sequence is 38, because the first cryptarithmically unique square is 38^2=1444. This means that no other square shares the same pattern "ABBB" of repeating digits.
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`%e Counterexample: 12 is not in this sequence because 12^2=144 has the same pattern "ABB" of repeating digits as 400=20^2. Equivalently, A358497(144)=A358497(400)=122.
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`%e 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 437 is a term in this sequence.
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`%t NumOfDigits = 4; (*Maximal integer length to be searched for*)
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`%t A358497[k_] := With[{pI = Values@PositionIndex@IntegerDigits@k}, MapIndexed[#1 -> Mod[#2[[1]], 10] &, pI, {2}] // Flatten // SparseArray // FromDigits];
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`%t Extract[Extract[Select[Tally[Table[{i, A358497[i^2]}, {i, 1, 10^NumOfDigits - 1}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}], {All, 1}]
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`%Y Cf. A374267 (cryptarithmically unique squares).
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`%K nonn,base
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`%O 1,1
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`%A _Dmytro Inosov_, Jul 02 2024
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