A275749 Prime numbers of the form 2*4^k - 27.

5, 101, 524261, 8388581

Offset: 1

Author

Timothy L. Tiffin, Aug 07 2016

Keywords

• nonn

Comments

Values of the exponent k are given in A275767, and every exponent (except for the first one) is odd. Consequently, after a(1) = 5, the rightmost digit of each term in this sequence will be 1.

As seen in the link below, a(5) = 2*4^291 - 27 > 3.1658 * 10^175. As a result of the recent extensions to A275767 by Vincenzo Librandi,

a(6) = 2*4^1263 - 27 > 5.0442 * 10^760

a(7) = 2*4^2661 - 27 > 2.4136 * 10^1602

a(8) = 2*4^3165 - 27 > 6.6206 * 10^1905

a(9) > 2*4^5000 - 27 > 3.9901 * 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) = 2*4^A275767(1) - 27 = 2*4^2  - 27 =      32 - 27 =       5.
a(2) = 2*4^A275767(2) - 27 = 2*4^3  - 27 =     128 - 27 =     101.
a(3) = 2*4^A275767(3) - 27 = 2*4^9  - 27 =  524288 - 27 =  524261.
a(4) = 2*4^A275767(4) - 27 = 2*4^11 - 27 = 8388608 - 27 = 8388581.
		

Links

• Timothy L. Tiffin, Table of n, a(n) for n = 1..5
• D. Alpern, Factorization using the Elliptic Curve Method.

Crossrefs

Cf. A275701, A275750, A275767.

Programs

Mathematica

Select[2*4^Range[2, 200] - 27, PrimeQ] (* Michael De Vlieger, Aug 08 2016 *)

Formula

a(n) = 2*4^A275767(n) - 27.