A347225 Lesser of twin primes (A001359) being both half-period primes (A097443).

197, 599, 881, 1277, 1997, 2081, 2237, 2801, 2999, 3359, 4721, 5279, 5879, 6197, 6959, 7877, 8837, 9239, 9719, 12161, 12239, 13721, 17921, 17957, 18521, 21839, 22637, 24917, 28277, 30557, 31319, 31721, 32117, 32441, 32717, 34757, 35081, 35279, 35837, 38921, 39239, 39839

Offset: 1

Author

Lamine Ngom, Aug 24 2021

Keywords

• nonn
• base

Comments

A proper subset of both A001359 and A097443.

Number of terms < 10^k: 0, 0, 3, 19, 86, 516, 3686, 27834, 216758, 1739358, …

A243096 provides lesser of twin primes being both full reptend primes (A001913, A006883): in other words, lesser of twin primes whose periods difference is 2.

This sequence lists lesser of twin primes whose periods difference is 1. Equivalently, these twin primes are both half-period primes (A097443).

The twin primes conjecture being true should imply that these two sequences are infinite.

Surprisingly, apart from 1 and 2, for any other value of k integer, it appears that the sequence "lesser of twin primes whose periods difference is k" is empty or contains at most two terms (no counterexample found for twin primes below 10^9).

Examples

			The decimal expansion 1/p for the prime p = 1277 has a periodic part length equal to (p-1)/2. 1277 is thus a half-period prime. The same applies for p + 2 = 1279 (prime). Hence 1277 is in sequence.
		

Crossrefs

Cf. A002371, A001359, A097443, A243096, A001913, A006883

Programs

Maple

select(t -> isprime(t) and isprime(t + 2) and numtheory:-order(10, t) = (t - 1)/2 and numtheory:-order(10, t + 2) = (t + 1)/2, [seq(t, t = 3 .. 40000, 2)]);

Formula

a(n) is congruent to {11, 17, 29} mod 30.