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a(n) = number of integers k <= n for which prime(k+1)-prime(k) is a multiple of three.
5

%I #12 Mar 20 2016 12:54:51

%S 0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,5,5,6,6,7,7,7,7,7,7,7,7,7,8,8,8,

%T 8,9,10,10,11,12,12,12,12,12,12,13,14,14,14,14,15,15,15,16,17,18,18,

%U 19,19,19,19,19,19,19,19,19,20,20,20,20,21,21,22,23,23,24,24,24,24,24,24,24,24,25,25,26,26,26,26,26,27

%N a(n) = number of integers k <= n for which prime(k+1)-prime(k) is a multiple of three.

%C a(n) = number of terms of A270190 <= n, the least monotonic left inverse of A270190.

%C See comments at A269364.

%H Antti Karttunen, <a href="/A269850/b269850.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(A270190(n)) = n.

%t Table[Count[Select[Range@ 125, Divisible[Prime[# + 1] - Prime@ #, 3] &], k_ /; k <= n], {n, 91}] (* _Michael De Vlieger_, Mar 17 2016 *)

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

%o (define A269850 (LEFTINV-LEASTMONO 1 1 A270190))

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

%Y Cf. A270190, A269849, A269364.

%K nonn

%O 1,11

%A _Antti Karttunen_, Mar 16 2016