A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 46, 47, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 91, 96, 97, 99, 100, 102, 103, 105, 107, 108, 110, 111, 114, 118, 119, 121, 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

Offset: 1

Author

Antti Karttunen, Mar 16 2016

Keywords

• nonn

Comments

Numbers n for which A001223(n) = 0 modulo 3.

See comments in A270189 and A269364.

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

Examples

			9 is present as the difference between A000040(9+1) = 29 and A000040(9) = 23 is 6, a multiple of three.
		

Links

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

Crossrefs

Complement: A270189.

Positions of zeros in A137264.

Left inverse: A269850.

Cf. also A001223, A269364, A270191, A270192, A269399.

Programs

Mathematica

Select[Range@ 210, Divisible[Prime[# + 1] - Prime@ #, 3] &] (* Michael De Vlieger, Mar 17 2016 *)
PrimePi/@Select[Partition[Prime[Range[350]],2,1],Divisible[#[[2]]-#[[1]], 3]&][[All,1]] (* Harvey P. Dale, Jul 11 2017 *)

PARI

isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ Michel Marcus, Mar 17 2016

Formula

Other identities. For all n >= 1:

a(n) = A269399(n) + 6.

A269850(a(n)) = n.