A298360 Numbers congruent to {3, 7, 13, 27} mod 30.

3, 7, 13, 27, 33, 37, 43, 57, 63, 67, 73, 87, 93, 97, 103, 117, 123, 127, 133, 147, 153, 157, 163, 177, 183, 187, 193, 207, 213, 217, 223, 237, 243, 247, 253, 267, 273, 277, 283, 297, 303, 307, 313, 327, 333, 337, 343, 357, 363, 367, 373, 387, 393, 397, 403

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 even 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 7 mod 10 if n is odd, or j is congruent to 3 mod 10 if n is even. 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*(3 + 4*x + 6*x^2 + 14*x^3 + 3*x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).

Example

37 belongs to this sequence and d = 37*2^16 + 1 is a divisor of F(9) = 2^(2^9) + 1, so 10 | (37 + (F(9)/d - 1)/2^16).

Mathematica

LinearRecurrence[{1, 0, 0, 1, -1}, {3, 7, 13, 27, 33}, 60]
CoefficientList[ Series[(3 + 4x + 6x^2 + 14x^3 + 3x^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..403] | n mod 30 in {3, 7, 13, 27}];
(PARI) Vec(x*(3 + 4*x + 6*x^2 + 14*x^3 + 3*x^4)/((1 + x)*(1 + x^2)*(1 - x)^2 + O(x^55)))

Crossrefs

Subsequence of A063226.

Cf. A000215, A299250.

Sequence in context: A125898 A146928 A068673 * A140465 A333653 A301594

Adjacent sequences: A298357 A298358 A298359 * A298361 A298362 A298363

Keyword

nonn,easy

Author

Arkadiusz Wesolowski, Feb 05 2018

Status

approved