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Numbers n such that 65 divides 4n^2 + 1; alternately, numbers which are 4, 9, 56, or 61 mod 65.
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%I #60 Mar 16 2023 08:50:19

%S 4,9,56,61,69,74,121,126,134,139,186,191,199,204,251,256,264,269,316,

%T 321,329,334,381,386,394,399,446,451,459,464,511,516,524,529,576,581,

%U 589,594,641,646,654,659,706,711,719,724,771,776,784,789,836,841,849

%N Numbers n such that 65 divides 4n^2 + 1; alternately, numbers which are 4, 9, 56, or 61 mod 65.

%C The sequence is infinite, since every number of the form 65*k + 56 is a member. - _Arkadiusz Wesolowski_, Oct 29 2013

%D Wacław Sierpiński, 200 zadan z elementarnej teorii liczb, Warsaw: PZWS, 1964, pp. 5, 29.

%H Arkadiusz Wesolowski, <a href="/A203464/b203464.txt">Table of n, a(n) for n = 1..10000</a>

%H W. Sierpiński, <a href="https://www.isinj.com/mt-usamo/250%20Problems%20in%20Elementary%20Number%20Theory%20-%20Sierpinski%20(1970).pdf">250 Problems in Elementary Number Theory</a>, (Modern Analytic and Computational Methods in Science and Mathematics, No. 26), American Elsevier Publishing Co., Inc., New York; PWN Polish Scientific Publishers, Warsaw, 1970, pp. 1, 23.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F From _Bruno Berselli_, Jan 12 2012: (Start)

%F G.f.: x*(4+5*x+47*x^2+5*x^3+4*x^4)/((1-x)^2*(1+x+x^2+x^3)).

%F a(n) = -a(-n+1) = (1/8)*(130*n+78*i^(n*(n+1))-45*(-1)^n-65), where i=sqrt(-1).

%F Sum(a(i), i=1..n) = a(A000982(n))+2*((-1)^n+1). (End)

%t Select[Range[850], Divisible[4*#^2 + 1, 65] &]

%t Flatten[Table[65*k + n, {k, 0, 12}, {n, {4, 9, 56, 61}}]]

%o (PARI) for(n=4, 850, if((4*n^2+1)%65==0, print1(n, ", ")))

%o (PARI) forstep(n=4,1e3,[5,47,5,8],print1(n", ")) \\ _Charles R Greathouse IV_, Jan 13 2012

%o (Magma) m:=54; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(4+5*x+47*x^2+5*x^3+4*x^4)/((1-x)^2*(1+x+x^2+x^3)))); // _Bruno Berselli_, Jan 12 2012

%o (Maxima) makelist((1/8)*(130*n+78*%i^(n*(n+1))-45*(-1)^n-65),n,1,53); /* _Bruno Berselli_, Jan 12 2012 */

%Y Cf. A053755.

%K nonn,easy

%O 1,1

%A _Arkadiusz Wesolowski_, Jan 06 2012