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.

-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, 20, 20, 20, 25, 25, 25, 9, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 9, 37, 38, 39, 40, 40, 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

Offset: 1

Author

Scott R. Shannon, Feb 02 2021

Keywords

• sign
• look
• base

Comments

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.

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.

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.

Examples

			a(5) = 5 as 5+5 = 10 which contains reverse(5-5) = reverse(0) = 0 as a substring.
a(6) = 5 as 6+5 = 11 which contains reverse(6-5) = reverse(1) = 1 as a substring.
a(15) = 10 as 15+10 = 25 which contains reverse(15-10) = reverse(5) = 5 as a substring.
a(22) = 9 as 22+9 = 31 which contains reverse(22-9) = reverse(13) = 31 as a substring.
		

Links

• Scott R. Shannon, Image of the terms for n = 5..300000. It is possible the frame-like pattern continues to larger and larger scales resulting in a fractal structure.

Crossrefs

Cf. A341034 (forward), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.