A051157 -id:A051157 - cp's OEIS frontend

A051154 a(n) = 1 + 2^k + 4^k where k = 3^n.

7, 73, 262657, 18014398643699713, 5846006549323611672814741748716771307882079584257

Offset: 0

The first three terms are prime. Are there more? Golomb shows that k must be a power of 3 in order for 1 + 2^k + 4^k to be prime. - T. D. Noe, Jul 16 2008

The next term, a(5) has 147 digits and is too large to include in DATA. - David A. Corneth, Aug 19 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..6
Dario Alpern, Factors of Generalized Fermat Numbers
Walter Feit, Finite projective planes and a question about primes, Proc. AMS, Vol. 108(1990), 561-564.
Solomon W. Golomb, Cyclotomic polynomials and factorization theorems, Amer. Math. Monthly 85 (1978), 734-737.

Cf. A001576, A051155, A051156, A051157.

F:= proc(n,r) local p; p := ithprime(r); (2^(p^(n+1))-1)/(2^(p^n)-1); end:
[ seq(F(n,2), n=0..5) ];

Table[4^(3^n) + 2^(3^n) + 1, {n, 1, 5}]  (* Artur Jasinski, Oct 31 2011 *)

a(n)=1+2^3^n+4^3^n \\ Charles R Greathouse IV, Oct 31 2011

a(n) = (2^(3^(n+1))-1)/(2^(3^n)-1).

A051156 a(n) = (2^p^2 - 1)/(2^p - 1) where p is the n-th prime.

Original entry on oeis.org

5, 73, 1082401, 4432676798593, 1298708349570020393652962442872833, 91355004067076339167413824240109498970069278721, 7588608256743087977590500540116743445925584618982806531428980886590618779326218241

Offset: 1

N. J. A. Sloane

Note that a(n) = Phi(p,2^p) or a(n) = Phi(p^2,2), where Phi(m,x) is the m-th cyclotomic polynomial and p is the n-th prime. - Thomas Ordowski, Feb 18 2014

Cf. A051154, A051155, A051157.

Table[Cyclotomic[Prime[n]^2, 2], {n, 7}] (* Arkadiusz Wesolowski, May 13 2012 *)
Table[(2^Prime[n]^2-1)/(2^Prime[n]-1),{n,10}] (* Harvey P. Dale, Apr 06 2019 *)

a(n) = A070526(prime(n)), a(n) = A019320(prime(n)^2). - Thomas Ordowski, Feb 18 2014

A051155 a(n) = (2^5^(n+1) - 1)/(2^5^n - 1).

Original entry on oeis.org

31, 1082401, 1267650638007162390353805312001, 3273390607896141870013189696827599152293599089395397756694773868291726792119530172040230983402733964346814858022765439290901496446006940490331586560001

Offset: 0

N. J. A. Sloane

Cf. A051154, A051156, A051157.

Table[(2^5^(n+1) - 1)/(2^5^n - 1),{n,0,3}] (* Stefano Spezia, Dec 23 2022 *)

cp's OEIS Frontend

A051154 a(n) = 1 + 2^k + 4^k where k = 3^n.

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A051156 a(n) = (2^p^2 - 1)/(2^p - 1) where p is the n-th prime.

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Formula

A051155 a(n) = (2^5^(n+1) - 1)/(2^5^n - 1).

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