A068227 - OEIS

%I #13 Apr 03 2023 10:36:10

%S 2,3,3,2,4,1,3,2,4,3,2,1,3,2,4,3,4,1,2,4,1,2,4,3,1,3,2,4,1,3,2,4,3,2,

%T 3,2,1,2,4,3,4,1,4,1,3,2,2,2,4,1,3,4,1,4,3,4,3,2,1,3,2,3,2,4,1,3,2,1,

%U 4,1,3,4,2,1,2,4,3,1,3,1,4,1,4,1,2,4,3,1,3,2,4,4,2,4,2,4,3,3,2,1,2,3,4

%N The "genity" sequence of the primes, i.e., a(n) = g(p) = ((p mod 4) + (p mod 6))/2, where p is the n-th prime.

%C The name "genity" was derived from "genes" and "parity", since the fourfold values of g(p) in a sequence corresponding to prime arguments resemble the genetic sequences of the nucleotides in the DNA. Parity is also related, since it originally means a (mod 2) feature, while here we categorize the primes (mod 4) and (mod 6), simultaneously.

%C The arithmetic function g(p) = ((p mod 4) + (p mod 6))/2 provides integer values for prime arguments, such that 1 <= g(p) <= 4 and is determined by the congruence class of p (mod 12). Specifically, g(p)=1 if p==1 (mod 12), g(p)=2 if p=2 or p==7 (mod 12), g(p)=3 if p=3 or p==5 (mod 12) and g(p)=4 if p==11 (mod 12).

%C Dickson's conjecture implies that every finite sequence of numbers from 1 to 4 occurs infinitely often in this sequence.

%H The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=DicksonsConjecture">Dickson's conjecture</a>

%t Table[(Mod[Prime[n], 4] + Mod[Prime[n], 6])/2, {n, 1, 100}]

%o (PARI) for(i=1,120,print((prime(i)%4+prime(i)%6)/2))

%Y Cf. A068228, A068229, A040117, A068231, A068232, A068233, A068234, A068235.

%K easy,nonn

%O 1,1

%A Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

%E Edited by _Dean Hickerson_ and _Robert G. Wilson v_, Mar 06 2002