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

%I A374267 #47 Aug 03 2024 19:15:03
%S A374267 1444,7744,14884,19881,29929,37636,40401,44944,46656,55696,66564,
%T A374267 69696,116964,133225,136161,144400,166464,190969,202500,219961,224676,
%U A374267 225625,261121,276676,277729,300304,339889,407044,438244,473344,511225,525625,544644,553536,555025,556516,585225
%N A374267 Perfect squares whose pattern of identical digits is unique among the squares.
%C A374267 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.
%C A374267 Each perfect square in this sequence has a unique digit pattern in the sense that no other square has the same pattern.
%C A374267 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.
%H A374267 Wikipedia, <a href="https://en.wikipedia.org/wiki/Verbal_arithmetic">Verbal arithmetic</a>.
%F A374267 a(n) = A374268(n)^2.
%e A374267 The first cryptarithmically unique square is 38^2=1444. This means that no other square has the same digit pattern "ABBB".
%e A374267 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.
%e A374267 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.
%t A374267 NumOfDigits = 4; (* Maximal integer length to be searched for *)
%t A374267 A358497[k_] := With[{pI = Values@PositionIndex@IntegerDigits@k}, MapIndexed[#1 -> Mod[#2[[1]], 10] &, pI, {2}] // Flatten // SparseArray // FromDigits];
%t A374267 Extract[Extract[Select[Tally[Table[{#, A358497[#]} &[i^2], {i, 1, 10^NumOfDigits - 1}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}], {All, 1}]
%Y A374267 Subsequence of A000290 (squares).
%Y A374267 Cf. A374268 (bases of cryptarithmically unique squares).
%Y A374267 Cf. A374238 (cryptarithmically unique primes).
%K A374267 nonn,base
%O A374267 1,1
%A A374267 _Dmytro Inosov_, Jul 02 2024