A051156 -id:A051156 - 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).

A128889 a(n) = (2^(n^2) - 1)/(2^n - 1).

Original entry on oeis.org

1, 5, 73, 4369, 1082401, 1090785345, 4432676798593, 72340172838076673, 4731607904558235517441, 1239150146850664126585242625, 1298708349570020393652962442872833, 5445847423328601499764522166702896582657, 91355004067076339167413824240109498970069278721

Offset: 1

Leroy Quet, Apr 19 2007

nonn

a(n) is prime for n in A156585. Conjecture: gpf(a(n)) = gpf(Phi(n,2^n)), where Phi(n,2^n) = A070526(n). - Thomas Ordowski, Feb 16 2014

The conjecture fails at n = 26, where 3340762283952395329506327023033 > 215656329382891550920192462661. Next counterexample for n = 30, but no odd counterexamples found so far. - Charles R Greathouse IV, Feb 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..50

Cf. A051156, A119408, A156585.

a:=n->(2^(n^2)-1)/(2^n-1): seq(a(n),n=1..13);

f[n_] := (2^(n^2) - 1)/(2^n - 1); Array[f, 12]
F[n_] := Plus @@ Table[2^((n - i)*n), {i, 1, n}] (* Enrique Pérez Herrero, Feb 23 2009 *)
Table[(2^(n^2) - 1)/(2^n - 1), {n, 1, 20}] (* Vincenzo Librandi, Feb 18 2014 *)

a(n)=(2^n^2-1)/(2^n-1) \\ Charles R Greathouse IV, Feb 17 2014

a(n) = Sum_{k=1..n} 2^((n-k)*n). - Enrique Pérez Herrero, Feb 23 2009

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 22 2007

A156585 Numbers such that (2^(n^2)-1)/(2^n-1) is prime.

Original entry on oeis.org

2, 3, 7, 59

Offset: 1

M. F. Hasler, Feb 10 2009

It is easy to see that all terms of this sequence must be prime; this motivates the definition of A051156(n) = (2^prime(n)^2-1)/(2^prime(n)-1).

No further terms up to n=1999. - Andreas Höglund, Apr 06 2018

Cf. A051156.

Select[Prime[Range[17]], PrimeQ[Cyclotomic[#^2, 2]] &] (* Arkadiusz Wesolowski, May 13 2012 *)

for/*prime*/( n=1,99, is/*pseudo*/prime( (2^n^2-1)/(2^n-1) ) & print1(n,","))

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 *)

A051157 a(n) = (2^p^3 - 1)/(2^p^2 - 1) where p = n-th prime.

Original entry on oeis.org

17, 262657, 1267650638007162390353805312001, 31828687130226401637050536789795514059715404495050094614691019248562308626412218127220737

Offset: 1

N. J. A. Sloane

Cf. A051154, A051155, A051156.

Table[(2^Prime[n]^3 - 1)/(2^Prime[n]^2 - 1),{n,4}] (* Stefano Spezia, Dec 23 2022 *)

A055515 a(n) = (2^n - 1)/product(2^p - 1) where the product is over all distinct primes p that divide n.

Original entry on oeis.org

1, 1, 1, 5, 1, 3, 1, 85, 73, 11, 1, 195, 1, 43, 151, 21845, 1, 12483, 1, 11275, 2359, 683, 1, 798915, 1082401, 2731, 19173961, 704555, 1, 1649373, 1, 1431655765, 599479, 43691, 8727391, 3272356035, 1, 174763, 9588151, 11822705675, 1, 1649061309, 1

Offset: 1

Leroy Quet, Jul 03 2000

			a(12) = (2^12 -1)/((2^2 -1) (2^3 -1)) = 195.

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

Cf. A055977.

Table[(2^n-1)/Times@@(2^#-1&/@FactorInteger[n][[;;,1]]),{n,50}] (* Harvey P. Dale, Jan 18 2025 *)

a(n) = my(f = factor(n)); (2^n-1)/prod(i=1, #f~, 2^f[i, 1] -1); \\ Michel Marcus, May 18 2014

For p prime, a(p) = 1. - Michel Marcus, May 18 2014

For p prime, a(p^2) = A051156(n). - Michel Marcus, May 18 2014

A065869 a(n) = (2^(prime(n)^2) + 1)/(2^prime(n) + 1).

Original entry on oeis.org

57, 1016801, 4363953127297, 1297440698667560637290197228189697, 91332703321545501540374299322682400349546864641, 7588492464653843921408409789152927606760448769729047438304822150910706202078740481

Offset: 2

Benoit Cloitre, Dec 07 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 2..16

Cf. A051156, A069255.

(2^#^2+1)/(2^#+1)&/@Prime[Range[2,10]] (* Harvey P. Dale, May 09 2025 *)

a(n) = { (2^(prime(n)^2) + 1)/(2^prime(n) + 1) } \\ Harry J. Smith, Nov 02 2009

a(n) = A069255((prime(n)+1)/2). - Andrew Howroyd, Dec 09 2024

A297625 Primes of the form (2^(p^k) - 1)/(2^(p^(k - 1)) - 1), with p prime and k > 1.

Original entry on oeis.org

5, 17, 73, 257, 65537, 262657, 4432676798593

Offset: 1

Arkadiusz Wesolowski, Jan 04 2018

nonn

Primes of the form Phi(x, 2), where x is a proper prime power and Phi is the cyclotomic polynomial.
Together with 3, supersequence of A019434.
Also called Mersenne-Fermat primes.
a(8) has 1031 digits and is too large to include.

Fredrick Kennard, Unsolved Problems in Mathematics, Lulu Press, 2015, p. 160.

Wikipedia, Mersenne-Fermat primes

Cf. A019434, A051156, A140797, A187823, A245730, A246547.

lst:=[]; r:=7; pr:=PrimesUpTo(r); for k in [2..r] do for c in [1..#pr] do p:=pr[c]; if p^k le r^2 then MF:=Truncate((2^(p^k)-1)/(2^(p^(k-1))-1)); if IsPrime(MF) then Append(~lst, MF); end if; end if; end for; end for; Sort(lst);

cp's OEIS Frontend

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

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A128889 a(n) = (2^(n^2) - 1)/(2^n - 1).

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A156585 Numbers such that (2^(n^2)-1)/(2^n-1) is prime.

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

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A051157 a(n) = (2^p^3 - 1)/(2^p^2 - 1) where p = n-th prime.

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A055515 a(n) = (2^n - 1)/product(2^p - 1) where the product is over all distinct primes p that divide n.

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A065869 a(n) = (2^(prime(n)^2) + 1)/(2^prime(n) + 1).

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A297625 Primes of the form (2^(p^k) - 1)/(2^(p^(k - 1)) - 1), with p prime and k > 1.

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