A001738 -id:A001738 - cp's OEIS frontend

A263608 Palindromes which are base-3 representations of squares.

0, 1, 11, 121, 10201, 11111, 112211, 122221, 1002001, 1120211, 11022011, 100020001, 101212101, 122111221, 1012112101, 1100220011, 10000200001, 10111011101, 110002200011, 111221122111, 1000002000001, 1001221221001, 1012200022101, 1101202021011, 1221221221221, 10101111110101

Offset: 1

N. J. A. Sloane, Oct 22 2015

Robert Israel, Table of n, a(n) for n = 1..143
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A002778, A003166, A029984, A262607, A029985.

Intersection of A001738 and A118594.

rev3:= proc(n) local L,i; L:= convert(n,base,3); add(L[-i]*3^(i-1),i=1..nops(L)) end proc:
c3:= proc(n) local L,i; L:= convert(n,base,3); add(L[i]*10^(i-1),i=1..nops(L)) end proc:
R:= 0,1: count:= 2:
for d from 2 while count < 100 do
    if d::odd then
      V:= select(issqr, [seq(seq(a*3^((d+1)/2) + b*3^((d-1)/2)+rev3(a),b=0..2),a=3^((d-3)/2) .. 3^((d-1)/2)-1)])
    else
      V:= select(issqr, [seq(a*3^(d/2) + rev3(a), a=3^(d/2-1) .. 3^(d/2)-1)]);
    fi;
    count:= count+nops(V);
    R:= R, op(map(c3,V));
od:
R; # Robert Israel, May 19 2024

Name edited by Robert Israel, May 19 2024

A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

Original entry on oeis.org

0, 0, 2, 1, 1, 2, 3, 2, 3, 2, 2, 4, 2, 4, 3, 3, 4, 4, 4, 4, 5, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 3, 3, 5, 3, 5, 3, 3, 5, 5, 5, 6, 4, 3, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 4, 5, 4, 4, 5, 5, 4, 5, 4, 5, 5, 5, 4, 6, 4, 5, 4, 5, 4, 5, 4, 4, 5, 4, 4, 4, 5, 5, 5, 4, 4, 6, 4, 5, 4, 4, 5, 5, 5, 5, 6

Offset: 0

Reinhard Zumkeller, Dec 18 2003

			a(12)=2: 12->'110', insert a 2 between the 1's and insert a 0 at the end: '12100'->144=12^2.

Robert Israel, Table of n, a(n) for n = 0..10000
Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A091092, A091091, A081604, A000290.

A091093:= proc(x) local L1, L2;
   L1:= convert(map(`+`,ListTools:-Reverse(convert(x,base,3)),48),bytes);
   L2:= convert(map(`+`,ListTools:-Reverse(convert(x^2,base,3)),48),bytes);
   StringTools:-Levenshtein(L1,L2)
end proc:
seq(A091093(i),i=0..1000); # Robert Israel, May 06 2014

a(n) = LevenshteinDistance(A007089(n), A001738(n)).

A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

Original entry on oeis.org

0, 1, 5, 6, 8, 11, 13, 16, 17, 18, 19, 33, 35, 38, 40, 56, 60, 74, 122, 123, 133, 143, 146, 164, 168, 173, 299, 350, 365, 429, 497, 515, 527, 564, 566, 593, 608, 611, 710, 1031, 1050, 1052, 1059, 1088, 1089, 1090, 1092, 1096, 1105, 1287, 1301, 1316, 1322

Offset: 1

Reinhard Zumkeller, Dec 18 2003

A091092(a(n)) = A091093(a(n)).

Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Binary
Eric Weisstein's World of Mathematics, Ternary

Cf. A007088, A001737, A007089, A001738, A000290.

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A263608 Palindromes which are base-3 representations of squares.

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A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

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A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

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