cp's OEIS Frontend

A275750 Prime numbers of the form 4^k - 27.

%I A275750 #37 Feb 27 2025 14:45:24
%S A275750 37,229,997,1048549,4194277,67108837,1125899906842597,
%T A275750 72057594037927909,288230376151711717,
%U A275750 1361129467683753853853498429727072845797,1393796574908163946345982392040522594123749,1725436586697640946858688965569256363112777243042596638790631055949797
%N A275750 Prime numbers of the form 4^k - 27.
%C A275750 Values of the exponent k 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 A275750 As a result of the recent extensions to A274519 by _Vincenzo Librandi_,
%C A275750 a(13) = 4^305  - 27 > 4.2491 * 10^183
%C A275750 a(14) = 4^515  - 27 > 1.1505 * 10^310
%C A275750 a(15) = 4^2029 - 27 > 3.7994 * 10^1221
%C A275750 a(16) = 4^2393 - 27 > 5.3648 * 10^1440
%C A275750 a(17) = 4^2605 - 27 > 2.3242 * 10^1568
%C A275750 a(18) = 4^3530 - 27 > 1.8696 * 10^2125
%C A275750 a(19) = 4^4036 - 27 > 8.2058 * 10^2429
%C A275750 a(20) = 4^4750 - 27 > 6.0947 * 10^2859
%C A275750 a(21) > 4^5000 - 27 > 1.9950 * 10^3010.
%C A275750 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 A275750 D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%F A275750 a(n) = 4^A274519(n) - 27.
%e A275750 a(1) = 4^A274519(1) - 27 = 4^3  - 27 =       64 - 27 =       37.
%e A275750 a(2) = 4^A274519(2) - 27 = 4^4  - 27 =      256 - 27 =      229.
%e A275750 a(3) = 4^A274519(3) - 27 = 4^5  - 27 =     1024 - 27 =      997.
%e A275750 a(4) = 4^A274519(4) - 27 = 4^10 - 27 =  1048576 - 27 =  1048549.
%e A275750 a(5) = 4^A274519(5) - 27 = 4^11 - 27 =  4194304 - 27 =  4194277.
%e A275750 a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.
%t A275750 Select[4^Range[3, 120] - 27, PrimeQ] (* _Michael De Vlieger_, Aug 08 2016 *)
%Y A275750 Cf. A274519, A275701, A275749.
%K A275750 nonn
%O A275750 1,1
%A A275750 _Timothy L. Tiffin_, Aug 07 2016