A267985 Numbers congruent to {7, 13} mod 30.

7, 13, 37, 43, 67, 73, 97, 103, 127, 133, 157, 163, 187, 193, 217, 223, 247, 253, 277, 283, 307, 313, 337, 343, 367, 373, 397, 403, 427, 433, 457, 463, 487, 493, 517, 523, 547, 553, 577, 583, 607, 613, 637, 643, 667, 673, 697, 703, 727, 733, 757, 763

Offset: 1

Author

Arkadiusz Wesolowski, Jan 23 2016

Keywords

• nonn
• easy

Comments

Union of A128471 and A082369.

For all k >= 1 the numbers 2^k - a(n) and a(n)*2^k - 1 do not form a pair of primes, where n is any positive integer.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A082369, A128471, A267984.

Programs

Magma

[n: n in [0..763] | n mod 30 in {7, 13}];

Mathematica

LinearRecurrence[{1, 1, -1}, {7, 13, 37}, 52]

PARI

Vec(x*(7 + 6*x + 17*x^2)/((1 + x)*(1 - x)^2) + O(x^53))

Formula

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.

G.f.: x*(7 + 6*x + 17*x^2)/((1 + x)*(1 - x)^2).

a(n) = a(n-2) + 30.

a(n) = 10*(3*n - 4) - a(n-1).

From Colin Barker, Jan 24 2016: (Start)

a(n) = (30*n-9*(-1)^n-25)/2 for n>0.

a(n) = 15*n-17 for n>0 and even.

a(n) = 15*n-8 for n odd.

(End)

E.g.f.: 17 + ((30*x - 25)*exp(x) - 9*exp(-x))/2. - David Lovler, Sep 10 2022

Extensions

Comment corrected by Philippe Deléham, Nov 28 2016