A191279 3-digit half-palindromes.

22, 51, 87, 91, 102, 121, 145, 169, 187, 190, 212, 220, 225, 245, 247, 248, 260, 287, 289, 290, 295, 345, 361, 364, 371, 425, 435, 441, 442, 445, 475, 477, 490, 495, 511, 529, 551, 574, 584, 603, 612, 625, 632, 651, 658, 663, 672, 679, 715, 721, 722, 728, 729

Offset: 1

Author

Vladimir Shevelev, May 29 2011

Keywords

• nonn
• base

Comments

A positive integer m we call k-digit half-palindrome if there exist two bases 1b=[m_k m(k-1)...m_1]_c, where m_i are digits in both of these bases with the condition m_1>0 and m_k>0 (see SeqFan Discussion list from Mar 03 2011, where we introduced "b,c-palindromes").

Robert Israel showed (see SeqFan Discussion list from the same day) that every number of the form [n+1,n,n]_(2*n+1)is 3-digit half-palindrome with b=2*n+1 and c=2*n+2. Thus the sequence is infinite.

On the other hand, every number of the form [k*n+m+1,0,k*n+m]_(4*k*n+4*m+1), where k>=1,m>=0, is 3-digit half-palindrome with b=4*k*n+4*m+1 and c=4*k*n+4*m+3.

Examples

			Let m=22. We have 22=[2 1 1]_3 and 22=[1 1 2]_4. Thus 22, by the definition, is a 3-digit half-palindrome.
Let m=91. We have 91=[3 3 1]_5 and 91 =[1 3 3]_8. Thus 91 is a 3-digit half palindrome.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1300

Crossrefs

Cf. A002113, A006995, A059809.

Programs

Mathematica

q[n_] := Module[{ans = False, db, dc}, Do[db = IntegerDigits[n, b]; If[Length[db] == 3, Do[dc = IntegerDigits[n, c]; If[Length[dc] == 3 && db == Reverse[dc], ans = True; Break[]], {c, b + 1, n - 1}]], {b, 2, n - 1}]; ans]; Select[Range[1000], q] (* Amiram Eldar, Jun 18 2025 *)

Extensions

Corrected by R. J. Mathar, Jul 02 2012

More terms from Amiram Eldar, Jun 18 2025