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%I #16 Jul 11 2017 17:15:45
%S 9,11,15,16,18,21,23,32,36,37,39,40,46,47,51,54,55,56,58,67,71,73,74,
%T 76,84,86,91,96,97,99,100,102,103,105,107,108,110,111,114,118,119,121,
%U 123,129,130,133,139,160,161,164,165,167,168,170,174,179,180,184,185,187,188,194,195,197,199,200,202,203,205,208,210
%N Numbers n for which prime(n+1)-prime(n) is a multiple of three.
%C Numbers n for which A001223(n) = 0 modulo 3.
%C See comments in A270189 and A269364.
%C Equivalently, numbers n for which prime(n+1)-prime(n) is a multiple of six. See A276414 for runs of increasing length of consecutive integers. - _M. F. Hasler_, Sep 03 2016
%H Antti Karttunen, <a href="/A270190/b270190.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 a(n) = A269399(n) + 6.
%F A269850(a(n)) = n.
%e 9 is present as the difference between A000040(9+1) = 29 and A000040(9) = 23 is 6, a multiple of three.
%t Select[Range@ 210, Divisible[Prime[# + 1] - Prime@ #, 3] &] (* _Michael De Vlieger_, Mar 17 2016 *)
%t PrimePi/@Select[Partition[Prime[Range[350]],2,1],Divisible[#[[2]]-#[[1]], 3]&][[All,1]] (* _Harvey P. Dale_, Jul 11 2017 *)
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A270190 (ZERO-POS 1 1 A137264))
%o (PARI) isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ _Michel Marcus_, Mar 17 2016
%Y Complement: A270189.
%Y Positions of zeros in A137264.
%Y Left inverse: A269850.
%Y Cf. also A001223, A269364, A270191, A270192, A269399.
%K nonn
%O 1,1
%A _Antti Karttunen_, Mar 16 2016