%I #54 Sep 27 2020 02:38:37
%S 36,169,1233,9745,77841,622609,4980753,39845905,318767121,2550136849,
%T 20401094673,163208757265,1305670058001,10445360463889,83562883710993,
%U 668503069687825,5348024557502481,42784196460019729,342273571680157713,2738188573441261585
%N a(n) = 19 * 8^n + 17 for n >= 0.
%C This sequence is the subject of the 4th problem of the 12th British Mathematical Olympiad in 1976 (see the link BMO).
%C Proposition: a(n) is never a prime number.
%C Proof:
%C If n is even, 3 divides a(n),
%C if n is odd with n = 4*k+1, 13 divides a(n), and
%C if n is odd with n = 4*k+3, 5 divides a(n).
%D A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 4 pp. 70 and 216-217 (1991).
%H Colin Barker, <a href="/A330770/b330770.txt">Table of n, a(n) for n = 0..1000</a>
%H British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo-1976.pdf">1976 - Problem 4</a>
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-8).
%F a(n) = 19 * A001018(n) + 17.
%F From _Colin Barker_, Feb 25 2020: (Start)
%F G.f.: (36 - 155*x) / ((1 - x)*(1 - 8*x)).
%F a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
%F (End)
%F E.g.f.: exp(x)*(17 + 19*exp(7*x)). - _Stefano Spezia_, Feb 25 2020
%e a(4) = 19 * 8^4 + 17 = 77841 = 3 * 25947.
%e a(5) = 19 * 8^5 + 17 = 622609 = 13 * 47893.
%e a(7) = 19 * 8^7 + 17 = 39845905 = 5 * 7969181.
%p B:=seq(19*8^n+17, n=0..40);
%t Table[19 * 8^n + 17, {n, 0, 19}] (* _Amiram Eldar_, Feb 23 2020 *)
%o (PARI) Vec((36 - 155*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ _Colin Barker_, Feb 25 2020
%Y Cf. A001018 (8^n).
%K nonn,easy
%O 0,1
%A _Bernard Schott_, Feb 23 2020
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