A299250 Numbers congruent to {9, 11, 21, 29} mod 30.

9, 11, 21, 29, 39, 41, 51, 59, 69, 71, 81, 89, 99, 101, 111, 119, 129, 131, 141, 149, 159, 161, 171, 179, 189, 191, 201, 209, 219, 221, 231, 239, 249, 251, 261, 269, 279, 281, 291, 299, 309, 311, 321, 329, 339, 341, 351, 359, 369, 371, 381, 389, 399, 401, 411

Offset

1,1

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 odd k >= m + 2 and n >= 1, then the cofactor of F(m) is equal to F(m)/b = j*2^k + 1, where j is congruent to 1 mod 10 if n == 0 or 1 mod 4, or j is congruent to 9 mod 10 if n == 2 or 3 mod 4. That is, the integer a(n) + j must be divisible by 10.

Links

Table of n, a(n) for n=1..55.

Wikipedia, Fermat number

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Formula

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

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

G.f.: x*(9 + 2*x + 10*x^2 + 8*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).

Example

39 belongs to this sequence and d = 39*2^13 + 1 is a divisor of F(11) = 2^(2^11) + 1, so 10 | (39 + (F(11)/d - 1)/2^13).

Mathematica

LinearRecurrence[{1, 0, 0, 1, -1}, {9, 11, 21, 29, 39}, 60]
CoefficientList[ Series[(9 + 2x + 10x^2 + 8x^3 + x^4)/((-1 + x)^2 (1 + x + x^2 + x^3)), {x, 0, 54}], x] (* Robert G. Wilson v, Feb 08 2018 *)

Prog

(Magma) [n: n in [0..411] | n mod 30 in {9, 11, 21, 29}];
(PARI) Vec(x*(9 + 2*x + 10*x^2 + 8*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2 + O(x^55)))

Crossrefs

Subsequence of A090771.

Cf. A000215, A298360.

Sequence in context: A195572 A046259 A258452 * A074345 A022323 A106525

Adjacent sequences: A299247 A299248 A299249 * A299251 A299252 A299253

Keyword

nonn,easy

Author

Arkadiusz Wesolowski, Feb 05 2018

Status

approved