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%I #14 Sep 08 2022 08:46:20
%S 9,11,21,29,39,41,51,59,69,71,81,89,99,101,111,119,129,131,141,149,
%T 159,161,171,179,189,191,201,209,219,221,231,239,249,251,261,269,279,
%U 281,291,299,309,311,321,329,339,341,351,359,369,371,381,389,399,401,411
%N Numbers congruent to {9, 11, 21, 29} mod 30.
%C 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.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F a(n) = a(n-4) + 30.
%F G.f.: x*(9 + 2*x + 10*x^2 + 8*x^3 + x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).
%e 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).
%t LinearRecurrence[{1, 0, 0, 1, -1}, {9, 11, 21, 29, 39}, 60]
%t 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 *)
%o (Magma) [n: n in [0..411] | n mod 30 in {9, 11, 21, 29}];
%o (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)))
%Y Subsequence of A090771.
%Y Cf. A000215, A298360.
%K nonn,easy
%O 1,1
%A _Arkadiusz Wesolowski_, Feb 05 2018
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