%I #32 Aug 11 2016 02:47:37
%S 37,229,997,1048549,4194277,67108837,1125899906842597,
%T 72057594037927909,288230376151711717,
%U 1361129467683753853853498429727072845797,1393796574908163946345982392040522594123749,1725436586697640946858688965569256363112777243042596638790631055949797
%N Prime numbers of the form 4^n - 27.
%C Values of the exponent n are given in A274519. If the exponent is odd, then the rightmost digit of a(n) will be 7. If the exponent is even, then the rightmost digit of a(n) will be 9.
%C As a result of the recent extensions to A274519 by _Vincenzo Librandi_,
%C a(13) = 4^305 - 27 > 4.2491 * 10^183
%C a(14) = 4^515 - 27 > 1.1505 * 10^310
%C a(15) = 4^2029 - 27 > 3.7994 * 10^1221
%C a(16) = 4^2393 - 27 > 5.3648 * 10^1440
%C a(17) = 4^2605 - 27 > 2.3242 * 10^1568
%C a(18) = 4^3530 - 27 > 1.8696 * 10^2125
%C a(19) = 4^4036 - 27 > 8.2058 * 10^2429
%C a(20) = 4^4750 - 27 > 6.0947 * 10^2859
%C a(21) > 4^5000 - 27 > 1.9950 * 10^3010.
%C These primes a(m) can be used to generate numbers having abundance 26. The formula a(m)*(a(m)+27)/2 produces some of the terms in A275701.
%H D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%F a(m) = 4^A274519(m) - 27.
%e a(1) = 4^A274519(1) - 27 = 4^3 - 27 = 64 - 27 = 37.
%e a(2) = 4^A274519(2) - 27 = 4^4 - 27 = 256 - 27 = 229.
%e a(3) = 4^A274519(3) - 27 = 4^5 - 27 = 1024 - 27 = 997.
%e a(4) = 4^A274519(4) - 27 = 4^10 - 27 = 1048576 - 27 = 1048549.
%e a(5) = 4^A274519(5) - 27 = 4^11 - 27 = 4194304 - 27 = 4194277.
%e a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.
%t Select[4^Range[3, 120] - 27, PrimeQ] (* _Michael De Vlieger_, Aug 08 2016 *)
%Y Cf. A274519, A275701, A275749.
%K nonn,more
%O 1,1
%A _Timothy L. Tiffin_, Aug 07 2016