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

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 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.

Examples

			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).
		

Links

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

Crossrefs

Subsequence of A090771.

Cf. A000215, A298360.

Programs

Magma

[n: n in [0..411] | n mod 30 in {9, 11, 21, 29}];

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 *)

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)))

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).