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a(n) = number of integers k <= n for which prime(k+1)-prime(k) is not a multiple of three.
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%I #15 Apr 15 2024 20:27:02

%S 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 20,21,22,23,24,24,25,26,27,27,27,28,28,28,29,30,31,32,33,33,33,34,35,

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

%C a(n) = number of terms of A270189 <= n, the least monotonic left inverse of A270189.

%C See comments at A269364.

%H Antti Karttunen, <a href="/A269849/b269849.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(A270189(n)) = n.

%t Table[Count[Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &], k_ /; k <= n], {n, 85}] (* _Michael De Vlieger_, Mar 17 2016 *)

%t Accumulate[If[Mod[#,3]==0,0,1]&/@Differences[Prime[Range[90]]]] (* _Harvey P. Dale_, Apr 15 2024 *)

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

%o (define A269849 (LEFTINV-LEASTMONO 1 1 A270189))

%o (PARI) a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0); \\ _Michel Marcus_, Mar 18 2016

%Y Cf. A270189, A269850, A269362, A269364.

%K nonn

%O 1,2

%A _Antti Karttunen_, Mar 16 2016