cp's OEIS Frontend

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

%I A298360 #36 Nov 15 2024 18:20:35
%S A298360 3,7,13,27,33,37,43,57,63,67,73,87,93,97,103,117,123,127,133,147,153,
%T A298360 157,163,177,183,187,193,207,213,217,223,237,243,247,253,267,273,277,
%U A298360 283,297,303,307,313,327,333,337,343,357,363,367,373,387,393,397,403
%N A298360 Numbers congruent to {3, 7, 13, 27} mod 30.
%C A298360 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.
%H A298360 Harvey P. Dale, <a href="/A298360/b298360.txt">Table of n, a(n) for n = 1..1000</a>
%H A298360 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>
%H A298360 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A298360 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F A298360 a(n) = a(n-4) + 30.
%F A298360 G.f.: x*(3 + 4*x + 6*x^2 + 14*x^3 + 3*x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).
%e A298360 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).
%t A298360 LinearRecurrence[{1, 0, 0, 1, -1}, {3, 7, 13, 27, 33}, 60]
%t A298360 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 *)
%t A298360 Select[Range[500],MemberQ[{3,7,13,27},Mod[#,30]]&] (* _Harvey P. Dale_, Nov 15 2024 *)
%o A298360 (Magma) [n: n in [0..403] | n mod 30 in {3, 7, 13, 27}];
%o A298360 (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)))
%Y A298360 Subsequence of A063226.
%Y A298360 Cf. A000215, A299250.
%K A298360 nonn,easy
%O A298360 1,1
%A A298360 _Arkadiusz Wesolowski_, Feb 05 2018