A216262 Primes p such that, for p < q < r three consecutive primes, p + 2q + 2r, 2p + q + 2r and 2p + 2q + r are all primes.

10193, 12113, 17683, 19501, 63743, 70793, 74317, 74797, 79657, 89231, 109073, 112657, 114371, 116993, 119237, 120431, 130211, 139801, 148573, 152123, 164881, 173867, 201623, 230017, 264919, 275543, 284927, 290761, 323537, 325643, 371873, 382777, 385193, 396061, 399403, 402817, 415201, 421273

Offset

1,1

Comments

From first 10^5 primes, only 92 are terms. Indices of primes are 1252, 1451, 2032,..., 95460, 97950, 98973.

Note that p == q == r (mod 6), e.g., {10193, 10211, 10223} == 5 mod 6 and {17683, 17707, 17713} == 1 mod 6.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

a(1) = p = 10193; s = {p, q, r} = {10193, 10211, 10223};
{{1,2,2}.s, {2,1,2}.s, {2,2,1}.s} = {51061, 51043, 51031} all primes.

Mathematica

pr=Partition[Prime[Range[40000]], 3, 1]; Reap[Do[s=pr[[k]]; If[Union[PrimeQ[{{1, 2, 2}.s, {2, 1, 2}.s, {2, 2, 1}.s}]]=={True}, Sow[Prime[k]]], {k, Length[pr]}]][[2, 1]]
tcpQ[{p_, q_, r_}]:=AllTrue[{p+2q+2r, 2p+q+2r, 2p+2q+r}, PrimeQ]; Select[ Partition[ Prime[Range[36000]], 3, 1], tcpQ][[All, 1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2017 *)

Crossrefs

Sequence in context: A128878 A050267 A102326 * A243410 A221119 A105582

Adjacent sequences: A216259 A216260 A216261 * A216263 A216264 A216265

Keyword

nonn

Author

Zak Seidov, Mar 15 2013

Status

approved