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

1,1

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Formula

Other identities. For all n >= 1:

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

A269850(a(n)) = n.

Example

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

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 *)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A270190 (ZERO-POS 1 1 A137264))
(PARI) isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ Michel Marcus, Mar 17 2016

Crossrefs

Complement: A270189.

Positions of zeros in A137264.

Left inverse: A269850.

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

Sequence in context: A101754 A348614 A113339 * A320701 A287340 A037009

Adjacent sequences: A270187 A270188 A270189 * A270191 A270192 A270193

Keyword

nonn

Author

Antti Karttunen, Mar 16 2016

Status

approved