%N Lexicographically earliest sequence with no duplicate term that produces only primes by the rounding technique explained in the Comments section.
%C See the assembly [a(n),a(n+1)] as a decimal number. Round this number to the closest integer. All rounded assemblies will produce a prime number.
%C "Rounding to the closest integer" is ambiguous for decimal numbers like (k.5) where k is an integer. Here we round such numbers to be rounded to k+1. The only occurrence of such a "rounding ambiguity" in the sequence happens with a(1) = 1 and a(2) = 5. Indeed, no more (k.5) "dilemmas" like that one will ever occur again as the integers 50, 500, 5000,... (that might produce together with the previous term k the decimal number k.50 or k.500 or k.5000...) cannot be part of the sequence; this is because 50, 500, 5000,... are not primes themselves (they end with 0) and neither are 51, 501, 5001,... (they are divisible by 3).
%H Jean-Marc Falcoz, <a href="/A307603/b307603.txt">Table of n, a(n) for n = 1..5001</a>
%e The sequence starts with 1,5,2,3,4,6,7,10,52,53,11,12,58,59,13,...
%e The assembly [a(1),a(2)] is 1.5 which rounded upwards produces 2;
%e The assembly [a(2),a(3)] is 5.2 which rounded to the closest integer produces 5;
%e The assembly [a(3),a(4)] is 2.3 which rounded to the closest integer produces 2;
%e The assembly [a(4),a(5)] is 3.4 which rounded to the closest integer produces 3;
%e The assembly [a(5),a(6)] is 4.6 which rounded to the closest integer produces 5;
%Y Cf. A173919 (Numbers that are prime or one less than a prime).
%A _Eric Angelini_ and _Jean-Marc Falcoz_, Apr 18 2019