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

Author

Arkadiusz Wesolowski, Feb 05 2018

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 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.

Examples

			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).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Wikipedia, Fermat number
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Subsequence of A063226.

Cf. A000215, A299250.

Programs

Magma

[n: n in [0..403] | n mod 30 in {3, 7, 13, 27}];

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 *)
Select[Range[500],MemberQ[{3,7,13,27},Mod[#,30]]&] (* Harvey P. Dale, Nov 15 2024 *)

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)))

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).