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

Author

Cino Hilliard, May 06 2007

Keywords

• nonn
• easy

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.

Examples

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

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).

Crossrefs

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

Programs

Magma

[30*n+1: n in [0..50]]; // Vincenzo Librandi, Jun 16 2011

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 *)

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

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