login
Palindromes m such that m - sum_of_digits(m) is also a palindrome.
0

%I #15 Jan 27 2025 21:11:41

%S 0,1,2,3,4,5,6,7,8,9,11,101,181,262,343,424,686,767,848,929,1001,

%T 10001,100001,1000001,10000001,100000001,1000000001,10000000001,

%U 100000000001,1000000000001

%N Palindromes m such that m - sum_of_digits(m) is also a palindrome.

%C It is conjectured that a(n) = 10^(n-18) + 1 for all n > 20. - _Derek Orr_, Apr 05 2015

%C Palindromes in the sequence A229621.

%e 767 - (7+6+7) = 747 (another palindrome). So, 767 is in this sequence.

%t palQ[n_]:=Module[{idn=IntegerDigits[n], idn2}, idn2=IntegerDigits[n - Total[idn]]; idn==Reverse[idn]&&idn2==Reverse[idn2]]; Select[Range[0, 2 10^6], palQ] (* _Vincenzo Librandi_, Apr 06 2015 *)

%o (Python)

%o def pal(n):

%o r = ''

%o for i in str(n):

%o r = i + r

%o return r == str(n)

%o def DS(n):

%o s = 0

%o for i in str(n):

%o s += int(i)

%o return s

%o {print(n, end=', ') for n in range(10**6) if pal(n) and pal(n-DS(n))}

%o ## Simplified by _Derek Orr_, Apr 05 2015

%o (PARI) b(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])} \\ Code from _David A. Corneth_ in A002113, Jun 06 2014

%o pal(n)=my(d=digits(n));Vecrev(d)==d

%o for(n=1,10^7,my(m=b(n), s=sumdigits(m));if(pal(m-s),print1(m,", "))) \\ _Derek Orr_, Apr 05 2015

%Y Cf. A229621, A002113.

%K nonn,base

%O 1,3

%A _Derek Orr_, Sep 26 2013