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A023109 a(0) = 0. For n > 0, smallest non-palindromic number k such that the smallest palindrome in the Reverse and Add! trajectory of k is reached after exactly n iterations.

%I A023109 #56 Feb 16 2025 08:32:34
%S A023109 0,10,19,59,69,166,79,188,193,1397,829,167,2069,1797,849,177,1496,739,
%T A023109 1798,10777,6999,1297,869,187,89,10797,10853,10921,10971,13297,10548,
%U A023109 13293,17793,20889,700269,106977,108933,80359,13697,10794,15891,1009227,1007619,1009246,1008628,600259,131996,70759,1007377,1001699,600279,141996,70269,10677,10833,10911
%N A023109 a(0) = 0. For n > 0, smallest non-palindromic number k such that the smallest palindrome in the Reverse and Add! trajectory of k is reached after exactly n iterations.
%C A023109 From _Felix Fröhlich_, May 28 2022: (Start)
%C A023109 Variant of A015994 not allowing palindromes as starting values.
%C A023109 Smallest non-palindromic k such that A033665(k) = n. (End)
%H A023109 Chai Wah Wu, <a href="/A023109/b023109.txt">Table of n, a(n) for n = 0..100</a>
%H A023109 Jason Doucette, <a href="http://www.jasondoucette.com/worldrecords.html#Most">World Records</a>
%H A023109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/196-Algorithm.html">196-Algorithm.</a>
%H A023109 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%t A023109 Table[ SelectFirst[Range[0, 20000], (np = #; i = 0;
%t A023109     While[ ! PalindromeQ[np] && i <= n, np = np + IntegerReverse[np];
%t A023109      i++]; i == n ) &] , {n, 0, 32}] (* _Robert Price_, Oct 16 2019 *)
%o A023109 (PARI)
%o A023109 rev(n)={d=digits(n);p="";for(i=1,#d,p=concat(Str(d[i]),p));return(eval(p))}
%o A023109 nbs(n)=if(n==rev(n),return(0));for(k=1,10^3,i=n+rev(n);if(rev(i)==i,return(k));n=i) \\ A033665
%o A023109 a(n)=for(k=1,10^8,if(nbs(k)==n,return(k)))
%o A023109 n=0;while(n<100,print1(a(n),", ");n++) \\ _Derek Orr_, Jul 28 2014
%o A023109 (Python)
%o A023109 def A023109(n):
%o A023109     if n > 0:
%o A023109         k = 0
%o A023109         while True:
%o A023109             m = k
%o A023109             for i in range(n):
%o A023109                 if str(m) == str(m)[::-1]:
%o A023109                     break
%o A023109                 m += int(str(m)[::-1])
%o A023109             else:
%o A023109                 if str(m) == str(m)[::-1]:
%o A023109                     return k
%o A023109             k += 1
%o A023109     else:
%o A023109         return 0
%o A023109 # _Chai Wah Wu_, Feb 08 2015
%o A023109 (PARI) revadd(n) = n+eval(concat(Vecrev(Str(n))))
%o A023109 iterationstosmallestpalindrome(n, bound) = my(x=n, i=0, d); while(1, if(i > bound, return(-1)); x=revadd(x); i++; d=digits(x); if(d==Vecrev(d), return(i)))
%o A023109 a(n) = if(n==0, return(0)); for(k=1, oo, my(d=digits(k)); if(d!=Vecrev(d), if(iterationstosmallestpalindrome(k, n)==n, return(k)))) \\ _Felix Fröhlich_, May 28 2022
%Y A023109 Cf. A015994, A033665, A006960.
%K A023109 nonn,base,nice
%O A023109 0,2
%A A023109 _David W. Wilson_
%E A023109 a(41)-a(55) verified and added by _Aldo González Lorenzo_, May 15 2011
%E A023109 Name edited by _Felix Fröhlich_, May 28 2022