A296716 Numbers congruent to {7, 11, 13, 29} mod 30.

7, 11, 13, 29, 37, 41, 43, 59, 67, 71, 73, 89, 97, 101, 103, 119, 127, 131, 133, 149, 157, 161, 163, 179, 187, 191, 193, 209, 217, 221, 223, 239, 247, 251, 253, 269, 277, 281, 283, 299, 307, 311, 313, 329, 337, 341, 343, 359, 367, 371, 373, 389, 397, 401, 403

Offset: 1

Author

Arkadiusz Wesolowski, Dec 19 2017

Keywords

• nonn
• easy

Comments

For any m >= 0, if F(m) = 2^(2^m) + 1 has a factor of the form b = a(n)*2^k + 1 with k >= m + 2 and n >= 1, then the integer F(m)/b is congruent to 13 or 19 mod 30.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A191062, A267985.

Programs

Magma

[n: n in [0..403] | n mod 30 in {7, 11, 13, 29}];

Mathematica

LinearRecurrence[{1, 0, 0, 1, -1}, {7, 11, 13, 29, 37}, 60]

PARI

Vec(x*(7 + 4*x + 2*x^2 + 16*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2 + O(x^55)))

Formula

a(n) = a(n-1) + a(n-4) - a(n-5), n >= 6.

a(n) = a(n-4) + 30.

G.f.: x*(7 + 4*x + 2*x^2 + 16*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).

a(n) = (-15 + 5*(-1)^n + (3+9*i)*(-i)^n + (3-9*i)*i^n + 30*n) / 4 where i=sqrt(-1). - Colin Barker, Dec 19 2017

E.g.f.: (5*(e^(-x) + (6*x - 3)*e^x) + 6*cos(x) + 18*sin(x))/4. - Iain Fox, Dec 19 2017