A140332 Products of two palindromes in base 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 54, 55, 56, 63, 64, 66, 72, 77, 81, 88, 99, 101, 110, 111, 121, 131, 132, 141, 151, 154, 161, 165, 171, 176, 181, 191, 198

Offset

1,3

Comments

Geneviève Paquin, p. 5: "Lemma 3.7: a Christoffel word can always be written as the product of two palindromes." Products of two palindromes in base 10 may be either a palindrome (e.g., 202 * 202 = 40804} or a nonpalindrome (e.g., 2 * 88 = 176, or 22 * 33 = 726}. Contains A115683 and A141322 as proper subsets.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Geneviève Paquin, On a generalization of Christoffel words: epichristoffel words, arXiv:0805.4174 [math.CO], 2008-2009.

Formula

{i*j such that i is in A002113 and j is in A002113} = A002113 UNION A115683.

Maple

digrev:= proc(n) local L, i; L:= convert(n, base, 10); add(L[-i]*10^(i-1), i=1..nops(L)) end:
N:=3:
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Palis:= [Res]:
Res:= 0:
for i from 2 to nops(Palis) while Palis[i]^2 <= 10^N do
  for j from i to nops(Palis) while Palis[i]*Palis[j] <= 10^N do
     Res:= Res, Palis[i]*Palis[j];
od od:sort(convert({Res}, list)); # Robert Israel, Jan 06 2020

Mathematica

pal = Select[ Range[0, 200], # == FromDigits@ Reverse@ IntegerDigits@ # &]; Select[ Union[ Times @@@ Tuples[pal, 2]], # <= 200 &] (* Giovanni Resta, Jun 20 2016 *)

Crossrefs

Cf. A002113, A115683, A141322.

Sequence in context: A084034 A084347 A051038 * A368955 A155182 A096076

Adjacent sequences: A140329 A140330 A140331 * A140333 A140334 A140335

Keyword

easy,nonn,base

Author

Jonathan Vos Post, May 28 2008

Extensions

Data corrected by Giovanni Resta, Jun 20 2016

Status

approved