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%I #38 Sep 08 2022 08:46:21
%S 14,234,1826,10770,55154,260274,1167026,5059890,21442994,89438514,
%T 368866226,1509026610,6137242034,24853275954,100327829426,
%U 404059098930,1624486948274,6522713868594,26165182536626,104883769004850,420204307937714,1682825158192434,6737324873467826
%N a(n) = 384*4^n - 576*3^n + 220*2^n - 14.
%C From _Bruno Berselli_, Jun 15 2018: (Start)
%C a(0) = 2*7 and a(40) = 2*232110255958477539427146457 are semiprimes. For which values of n > 40 is a(n) semiprime?
%C For odd n, a(n) is divisible by 2*3.
%C For n == 3 (mod 4), a(n) is divisible by 2*3*5.
%C For n == 0 or 5 (mod 6), a(n) is divisible by 2*7.
%C For n == 2 or 4 (mod 5), a(n) is divisible by 2*11.
%C For n == 1 or 11 (mod 12), a(n) is divisible by 2*3*13.
%C For n == 15 (mod 16), a(n) is divisible by 2*3*5*17^2, etc.
%C If a(n) is divisible by 37 then it is also divisible by 3*5*7*13*19*73. (End)
%H Takao Komatsu, <a href="https://arxiv.org/abs/1806.05515">On poly-Euler numbers of the second kind</a>, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F G.f.: 2*(7 + 47*x - 12*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)).
%F a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).
%F a(n) = 14*A000453(n+4) + 94*A000453(n+3) - 24*A000453(n+2) for n>1. - _Bruno Berselli_, Jun 15 2018
%t Table[384 4^n - 576 3^n + 220 2^n - 14, {n, 0, 30}]
%o (Magma) [384*4^n-576*3^n+220*2^n-14: n in [0..30]];
%o (PARI) a(n) = 384*4^n - 576*3^n + 220*2^n - 14; \\ _Michel Marcus_, Jul 03 2018
%Y Cf. A000453, A000918, A305861.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, Jun 15 2018
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