A374268 Numbers whose squares have a unique pattern of identical digits among the squares.

38, 88, 122, 141, 173, 194, 201, 212, 216, 236, 258, 264, 342, 365, 369, 380, 408, 437, 450, 469, 474, 475, 511, 526, 527, 548, 583, 638, 662, 688, 715, 725, 738, 744, 745, 746, 765, 796, 804, 813, 816, 836, 880, 893, 898, 908, 970, 995, 1020

Offset: 1

Author

Dmytro Inosov, Jul 02 2024

Keywords

• nonn
• base

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.

The square of each term 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 of numbers in this sequence are unique solutions, so we call them cryptarithmically unique.

Examples

			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.
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.
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.
		

Links

• Wikipedia, Verbal arithmetic.

Crossrefs

Cf. A374267 (cryptarithmically unique squares).

Programs

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[{i, A358497[i^2]}, {i, 1, 10^NumOfDigits - 1}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}], {All, 1}]

Formula

a(n) = sqrt(A374267(n)).