A347225 - OEIS

%I #10 Sep 04 2021 20:35:52

%S 197,599,881,1277,1997,2081,2237,2801,2999,3359,4721,5279,5879,6197,

%T 6959,7877,8837,9239,9719,12161,12239,13721,17921,17957,18521,21839,

%U 22637,24917,28277,30557,31319,31721,32117,32441,32717,34757,35081,35279,35837,38921,39239,39839

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

%C A proper subset of both A001359 and A097443.

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

%C 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.

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

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

%C 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).

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

%e 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.

%p 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)]);

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

%K nonn,base

%O 1,1

%A _Lamine Ngom_, Aug 24 2021