|
%I #14 Mar 20 2016 12:53:01
%S 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,
%T 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,
%U 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
%N Prime number gaps read modulo 3.
%C 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.
%C 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
%C See comments at A269364. - _Antti Karttunen_, Mar 17 2016
%H Terence Tao, <a href="https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/">Biases between consecutive primes</a>, blog entry March 14, 2016
%t n=1000;(*The length of the list*) Mod[Differences[Table[Prime[i], {i, n}]], 3]
%o (Scheme) (define (A137264 n) (modulo (A001223 n) 3)) ;; _Antti Karttunen_, Mar 16 2016
%Y Cf. A001223.
%Y Cf. also A269364, A270189, A270190, A270191, A270192.
%K easy,nonn
%O 1,2
%A Noel H. Patson (n.patson(AT)cqu.edu.au), Mar 12 2008
|
