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A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.

%I A341028 #25 Jan 19 2024 23:54:58
%S A341028 -1,-1,-1,-1,5,5,5,5,5,5,5,5,5,5,10,10,10,10,10,15,15,9,15,15,20,20,
%T A341028 20,20,20,25,25,25,9,25,29,30,30,30,30,33,34,35,35,9,37,38,39,40,40,
%U A341028 41,42,43,44,45,9,50,50,50,50,55,41,51,52,53,54,9,60,60,60,65,50,32,52,53,54,70,9
%N A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.
%C A341028 Based on a search limit of 5*10^9 up to n = 300000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which a(n) = -1.
%C A341028 The longest run of consecutive terms with the same value in the first 300000 terms is the run of 5's at the beginning of the sequence, ten in all. This is likely the longest run for all numbers.
%C A341028 Numerous patterns exist in the values of a(n), e.g., when a(n) consists of all 9's and n is not a power of 10 then n is palindromic.
%H A341028 Scott R. Shannon, <a href="/A341028/a341028_2.png">Image of the terms for n = 5..300000</a>. It is possible the frame-like pattern continues to larger and larger scales resulting in a fractal structure.
%e A341028 a(5) = 5 as 5+5 = 10 which contains reverse(5-5) = reverse(0) = 0 as a substring.
%e A341028 a(6) = 5 as 6+5 = 11 which contains reverse(6-5) = reverse(1) = 1 as a substring.
%e A341028 a(15) = 10 as 15+10 = 25 which contains reverse(15-10) = reverse(5) = 5 as a substring.
%e A341028 a(22) = 9 as 22+9 = 31 which contains reverse(22-9) = reverse(13) = 31 as a substring.
%Y A341028 Cf. A341034 (forward), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.
%K A341028 sign,look,base
%O A341028 1,5
%A A341028 _Scott R. Shannon_, Feb 02 2021