A307603 Lexicographically earliest sequence with no duplicate term that produces only primes by the rounding technique explained in the Comments section.

1, 5, 2, 3, 4, 6, 7, 10, 52, 53, 11, 12, 58, 59, 13, 16, 60, 61, 17, 18, 66, 67, 19, 22, 70, 71, 23, 28, 72, 73, 29, 30, 78, 79, 31, 36, 82, 83, 37, 40, 88, 89, 41, 42, 96, 97, 43, 46, 502, 503, 47, 100, 508, 509, 101, 102, 520, 521, 103, 106, 522, 523, 107, 108, 540, 541, 109, 112, 546, 547, 113, 126, 556, 557, 127, 130, 562

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 18 2019

Keywords

• base
• nonn

Comments

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.

"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).

Examples

			The sequence starts with 1,5,2,3,4,6,7,10,52,53,11,12,58,59,13,...
The assembly [a(1),a(2)] is 1.5 which rounded upwards produces 2;
The assembly [a(2),a(3)] is 5.2 which rounded to the closest integer produces 5;
The assembly [a(3),a(4)] is 2.3 which rounded to the closest integer produces 2;
The assembly [a(4),a(5)] is 3.4 which rounded to the closest integer produces 3;
The assembly [a(5),a(6)] is 4.6 which rounded to the closest integer produces 5;
etc.
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..5001

Crossrefs

Cf. A173919 (Numbers that are prime or one less than a prime).