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Search: id:A068227
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| A068227 |
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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. |
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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%4+p%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.
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LINKS
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The Prime Glossary, Dickson's conjecture
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MATHEMATICA
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Table[(Mod[Prime[n], 4] + Mod[Prime[n], 6])/2, {n, 1, 100}]
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PROGRAM
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(PARI) for(i=1, 120, print((prime(i)%4+prime(i)%6)/2))
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CROSSREFS
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Cf. A068228, A068229, A040117, A068231, A068232, A068233, A068234, A068235.
Sequence in context: A141744 A089783 A090414 this_sequence A046824 A130156 A139169
Adjacent sequences: A068224 A068225 A068226 this_sequence A068228 A068229 A068230
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KEYWORD
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easy,nonn
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AUTHOR
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Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 06 2002
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