A137264 Prime number gaps read modulo 3.

1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 1, 2, 1, 0, 2, 1, 2, 0, 0, 1, 0, 0, 2, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 0, 2, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0

Offset: 1

Author

Noel H. Patson (n.patson(AT)cqu.edu.au), Mar 12 2008

Keywords

• easy
• nonn

Comments

Conjecture: The only digit that is repeated in the sequence is 0 except for n=2 and n=3 where 2 repeats. So 1 may be followed by 2 or 0; 2 may be followed by 1 or 0; 0 may be followed by 0 or 1 or 2. this has been confirmed for the first million prime gaps.

The conjecture is true, because any three numbers whose differences are (1, 1) or (2, 2) will form a complete residue system modulo 3, and hence one of them will be a multiple of 3. - Karl W. Heuer, Mar 16 2016

See comments at A269364. - Antti Karttunen, Mar 17 2016

Links

• Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Crossrefs

Cf. A001223.

Cf. also A269364, A270189, A270190, A270191, A270192.

Programs

Mathematica

n=1000;(*The length of the list*) Mod[Differences[Table[Prime[i], {i, n}]], 3]

Scheme

(define (A137264 n) (modulo (A001223 n) 3)) ;; Antti Karttunen, Mar 16 2016