A128470 a(n) = 30*n + 1.

1, 31, 61, 91, 121, 151, 181, 211, 241, 271, 301, 331, 361, 391, 421, 451, 481, 511, 541, 571, 601, 631, 661, 691, 721, 751, 781, 811, 841, 871, 901, 931, 961, 991, 1021, 1051, 1081, 1111, 1141, 1171, 1201, 1231, 1261, 1291, 1321, 1351, 1381, 1411, 1441, 1471

Offset

0,2

Comments

Possible upper bounds of twin primes pairs ending in 1: For a 30k + r "wheel", k > 0, r = 1, 13, 19 are the only possible values that can form an upper bound of a twin prime pair. The 30k+r wheel gives the sequence 1, 7, 11, 13, 17, 19, 23, 29 31, 37, 41, 43, 47, 49, 53, 59, ... which is frequently used in prime number sieves to skip multiples of 2, 3, 5. The fact that subtracting 2 from 30k+7, 11, 17, 23 will give us a multiple of 3 or 5 precludes these numbers from being an upper bound of a twin prime pair. This leaves us with r = 1, 13, 19 for k > 0 as the only possible cases to form an upper bound of a twin prime pair. 1, 13, 19 concludes the 6 numbers of the 8 number wheel that can form part of a twin prime pair.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Albert van der Horst, Counting Twin Primes

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 2*a(n-1) - a(n-2) for n > 1. - Vincenzo Librandi, Dec 30 2014

G.f.: (1 + 29*x)/(1 - x)^2. - Vincenzo Librandi, Dec 30 2014

E.g.f.: (1 + 30*x)*exp(x). - G. C. Greubel, Apr 04 2016

Example

61 = 30 * 2 + 1, the upper part of the twin prime pair 59, 61.

Mathematica

Range[1, 3001, 30] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
CoefficientList[Series[(1 + 29 x) / (1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 30 2014 *)
LinearRecurrence[{2, -1}, {1, 31}, 100] (* G. C. Greubel, Apr 04 2016 *)

Prog

(Magma) [30*n+1: n in [0..50]]; // Vincenzo Librandi, Jun 16 2011
(PARI) a(n)=30*n+1 \\ Charles R Greathouse IV, Oct 07 2015
(Scala) (0 to 49).map(30 * _ + 1) // Alonso del Arte, Jun 02 2019

Crossrefs

Cf. A161700, A005408, A016813, A016921, A017281, A017533, A158057, A161705, A161709, A161714.

Sequence in context: A073650 A078562 A054804 * A195744 A331324 A326896

Adjacent sequences: A128467 A128468 A128469 * A128471 A128472 A128473

Keyword

nonn,easy

Author

Cino Hilliard, May 06 2007

Status

approved