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

37, 229, 997, 1048549, 4194277, 67108837, 1125899906842597, 72057594037927909, 288230376151711717, 1361129467683753853853498429727072845797, 1393796574908163946345982392040522594123749, 1725436586697640946858688965569256363112777243042596638790631055949797

Offset: 1

Author

Timothy L. Tiffin, Aug 07 2016

Keywords

• nonn

Comments

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.

As a result of the recent extensions to A274519 by Vincenzo Librandi,

a(13) = 4^305 - 27 > 4.2491 * 10^183

a(14) = 4^515 - 27 > 1.1505 * 10^310

a(15) = 4^2029 - 27 > 3.7994 * 10^1221

a(16) = 4^2393 - 27 > 5.3648 * 10^1440

a(17) = 4^2605 - 27 > 2.3242 * 10^1568

a(18) = 4^3530 - 27 > 1.8696 * 10^2125

a(19) = 4^4036 - 27 > 8.2058 * 10^2429

a(20) = 4^4750 - 27 > 6.0947 * 10^2859

a(21) > 4^5000 - 27 > 1.9950 * 10^3010.

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.

Examples

			a(1) = 4^A274519(1) - 27 = 4^3  - 27 =       64 - 27 =       37.
a(2) = 4^A274519(2) - 27 = 4^4  - 27 =      256 - 27 =      229.
a(3) = 4^A274519(3) - 27 = 4^5  - 27 =     1024 - 27 =      997.
a(4) = 4^A274519(4) - 27 = 4^10 - 27 =  1048576 - 27 =  1048549.
a(5) = 4^A274519(5) - 27 = 4^11 - 27 =  4194304 - 27 =  4194277.
a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.
		

Links

• D. Alpern, Factorization using the Elliptic Curve Method.

Crossrefs

Cf. A274519, A275701, A275749.

Programs

Mathematica

Select[4^Range[3, 120] - 27, PrimeQ] (* Michael De Vlieger, Aug 08 2016 *)

Formula

a(n) = 4^A274519(n) - 27.