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

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

Offset

1,1

Comments

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.

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.

Links

Table of n, a(n) for n=1..12.

D. Alpern, Factorization using the Elliptic Curve Method.

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A274519, A275701, A275749.

Sequence in context: A133958 A088544 A051463 * A142445 A155974 A190603

Adjacent sequences: A275747 A275748 A275749 * A275751 A275752 A275753

Keyword

nonn,more

Author

Timothy L. Tiffin, Aug 07 2016

Status

approved