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

1,1

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).

Links

Table of n, a(n) for n=1..42.

Formula

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

Example

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.

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)]);

Crossrefs

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

Sequence in context: A142230 A142414 A142812 * A204844 A031706 A201256

Adjacent sequences: A347222 A347223 A347224 * A347226 A347227 A347228

Keyword

nonn,base

Author

Lamine Ngom, Aug 24 2021

Status

approved