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Numbers n for which (prime(n+1)-prime(n)) is not a multiple of three.
10

%I #17 Mar 20 2016 12:55:31

%S 1,2,3,4,5,6,7,8,10,12,13,14,17,19,20,22,24,25,26,27,28,29,30,31,33,

%T 34,35,38,41,42,43,44,45,48,49,50,52,53,57,59,60,61,62,63,64,65,66,68,

%U 69,70,72,75,77,78,79,80,81,82,83,85,87,88,89,90,92,93,94,95,98,101,104,106,109,112,113,115,116,117,120,122,124,125

%N Numbers n for which (prime(n+1)-prime(n)) is not a multiple of three.

%C See A269364 for the effect of the bias that favors these terms over the terms of A270190.

%H Antti Karttunen, <a href="/A270189/b270189.txt">Table of n, a(n) for n = 1..10000</a>

%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

%F Other identities. For all n >= 1:

%F A269849(a(n)) = n.

%t Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &] (* _Michael De Vlieger_, Mar 17 2016 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A270189 (NONZERO-POS 1 1 A137264))

%o (PARI) isok(n) = ((prime(n+1) - prime(n)) % 3) != 0; \\ _Michel Marcus_, Mar 17 2016

%Y Complement: A270190.

%Y Disjoint union of A270191 and A270192.

%Y Positions of 1's and 2's in A137264.

%Y Left inverse: A269849.

%Y Cf. A000040, A001223, A269364, A269389.

%K nonn

%O 1,2

%A _Antti Karttunen_, Mar 16 2016