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A118558 a(n) = (2^n-1)^4 - 2. 0

%I #15 Feb 19 2016 08:31:38

%S -2,-1,79,2399,50623,923519,15752959,260144639,4228250623,68184176639,

%T 1095222947839,17557851463679,281200199450623,4501401006735359,

%U 72040003462430719,1152780773560811519,18445618199572250623,295138898083176775679,4722294425687923097599

%N a(n) = (2^n-1)^4 - 2.

%C Exponent-4 analog of what for exponent 2 is A093112 (2^n-1)^2 - 2 = 4^n - 2^{n+1} - 1 and exponent 3 is A098878 (2^n - 1)^3 - 2. Primes include a(n) for n = 0, 2, 3, 11, 57; a type of "near-biquadratic primes." No more primes through (2^100-1)^4 - 2. Semiprimes include a(n) for n = 5, 6, 8, 10, 13, 14, 19, 20, 21, 25, 33, 35, 36, 40, 43, 51, 53, 63.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime.</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (31,-310,1240,-1984,1024).

%F a(n) = (2^n - 1)^4 - 2.

%F G.f.: x*(1984*x^4-2120*x^3+510*x^2-61*x+2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). - _Colin Barker_, Apr 30 2013

%e a(0) = (2^0 - 1)^4 - 2 = 0^4 - 2 = -2.

%e a(1) = (2^1 - 1)^4 - 2 = 1^4 - 2 = -1.

%e a(2) = (2^2 - 1)^4 - 2 = 3^4 - 2 = 79 (prime).

%e a(3) = (2^3 - 1)^4 - 2 = 7^4 - 2 = 2399 (prime).

%e a(11) = (2^11 - 1)^4 - 2 = 17557851463679 (prime).

%e a(57) = (2^57 - 1)^4 - 2 = 431359146674410224742050828377557509468732765984721170947417969786879 (prime).

%o (PARI) a(n)=(2^n-1)^4-2 \\ _Charles R Greathouse IV_, Feb 19 2016

%Y Cf. A091516, A091515, A098878, A091514.

%K easy,sign

%O 1,1

%A _Jonathan Vos Post_, May 03 2006

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)