OFFSET
1,1
COMMENTS
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.
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).
Dickson's conjecture implies that every finite sequence of numbers from 1 to 4 occurs infinitely often in this sequence.
LINKS
The Prime Glossary, Dickson's conjecture
MATHEMATICA
Table[(Mod[Prime[n], 4] + Mod[Prime[n], 6])/2, {n, 1, 100}]
PROG
(PARI) for(i=1, 120, print((prime(i)%4+prime(i)%6)/2))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002
EXTENSIONS
Edited by Dean Hickerson and Robert G. Wilson v, Mar 06 2002
STATUS
approved