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A299250 Numbers congruent to {9, 11, 21, 29} mod 30. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 17 11:26 EDT 2019. Contains 328108 sequences. (Running on oeis4.)