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

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

Offset: 1

Author

Antti Karttunen, Mar 16 2016

Keywords

• nonn

Comments

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

See comments at A269364.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Crossrefs

Cf. A270189, A269850, A269362, A269364.

Programs

Mathematica

Table[Count[Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &], k_ /; k <= n], {n, 85}] (* Michael De Vlieger, Mar 17 2016 *)
Accumulate[If[Mod[#,3]==0,0,1]&/@Differences[Prime[Range[90]]]] (* Harvey P. Dale, Apr 15 2024 *)

PARI

a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0); \\ Michel Marcus, Mar 18 2016

Formula

Other identities. For all n >= 1:

a(A270189(n)) = n.