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A269849 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is not a multiple of three.

%I A269849 #15 Apr 15 2024 20:27:02
%S A269849 1,2,3,4,5,6,7,8,8,9,9,10,11,12,12,12,13,13,14,15,15,16,16,17,18,19,
%T A269849 20,21,22,23,24,24,25,26,27,27,27,28,28,28,29,30,31,32,33,33,33,34,35,
%U A269849 36,36,37,38,38,38,38,39,39,40,41,42,43,44,45,46,47,47,48,49,50,50,51,51,51,52,52,53,54,55,56,57,58,59,59,60
%N A269849 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is not a multiple of three.
%C A269849 a(n) = number of terms of A270189 <= n, the least monotonic left inverse of A270189.
%C A269849 See comments at A269364.
%H A269849 Antti Karttunen, <a href="/A269849/b269849.txt">Table of n, a(n) for n = 1..10000</a>
%H A269849 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 A269849 Other identities. For all n >= 1:
%F A269849 a(A270189(n)) = n.
%t A269849 Table[Count[Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &], k_ /; k <= n], {n, 85}] (* _Michael De Vlieger_, Mar 17 2016 *)
%t A269849 Accumulate[If[Mod[#,3]==0,0,1]&/@Differences[Prime[Range[90]]]] (* _Harvey P. Dale_, Apr 15 2024 *)
%o A269849 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A269849 (define A269849 (LEFTINV-LEASTMONO 1 1 A270189))
%o A269849 (PARI) a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0); \\ _Michel Marcus_, Mar 18 2016
%Y A269849 Cf. A270189, A269850, A269362, A269364.
%K A269849 nonn
%O A269849 1,2
%A A269849 _Antti Karttunen_, Mar 16 2016