A156585 -id:A156585 - cp's OEIS frontend

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

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

A245730 Primes of the form 1+2^k+2^(2k)+...+2^((j-1)k) for some k>0, j>0.

Original entry on oeis.org

3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, 2147483647, 4432676798593, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Chai Wah Wu, Jul 30 2014

nonn

Contains the Mersenne primes A000668 which correspond to k=1. In base 2, primes with n 1's and k-1 0's between pairs of 1's.  Is a factor of 2^(n*k)-1.
Primes of the form (2^(n*k)-1)/(2^k-1). k=1 gives Mersenne primes 2^n-1 for n in A000043. n=2 and k=2^m gives Fermat primes 2^(2^m)+1 (A019434) for m = 0 to 4. k=n gives (2^(n^2)-1)/(2^n-1) which is prime for n = 2, 3, 7, 59 (A156585, n must be prime). The only other term below 2000 digits is 262657 for k=9 and n=3. - Jens Kruse Andersen, Aug 02 2014
The case n=3 gives the primes in A051154. - John Blythe Dobson
Wells Johnson (1977), 199, Corollary 6, proved that members of this sequence cannot be Wieferich primes (A001220). - John Blythe Dobson

			The number 4432676798593 is in the list as it is prime and it is equal to 1+2^7+2^(2*7)+2^(3*7)+2^(4*7)+2^(5*7)+2^(6*7).

Wells Johnson, On the nonvanishing of Fermat quotients (mod p), J. für Math. 292 (1977), 196-200.

Jens Kruse Andersen, Table of n, a(n) for n = 1..25

Cf. A000043, A000668, A019434, A156585.

from sympy import isprime
sorted([int(('0'*m+'1')*n,2) for m in range(50) for n in range(1,50) if isprime(int(('0'*m+'1')*n,2))])

A006514 Primes p such that 2^p - 1 has at most 2 prime factors.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 59, 61, 67, 83, 89, 97, 101, 103, 107, 109, 127, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 521, 523, 607, 727, 809, 881, 971, 983, 997, 1061, 1063

Offset: 1

N. J. A. Sloane

nonn

For a composite n, number 2^n - 1 has at most 2 prime factors only if n = p^2, where p is prime from the intersection of A000043 and A156585. The only known such primes are 2, 3, 7. - Max Alekseyev, Apr 23 2019

a(54) >= 1277. - Max Alekseyev, Apr 23 2019

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000043 (a subsequence), A001348, A088863.

Select[Prime[Range[100]],PrimeOmega[2^#-1]<3&] (* Harvey P. Dale, Nov 11 2011 *)

More terms from Sean A. Irvine, May 04 2017

Edited by Max Alekseyev, Apr 23 2019

A353101 Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.

Original entry on oeis.org

2, 2, 22, 2, 43, 24, 315, 38, 54, 265, 605, 61, 697, 306, 1153, 370, 2, 10688, 3075, 2338, 1153, 3243, 130, 2301, 315, 200, 1155, 14739, 4591, 2230, 263, 6665, 250, 10520, 2228, 3699, 1126, 8925, 8732, 10556, 19860, 29121, 32804, 4666, 2313, 27398, 14280, 2013, 29022, 26131, 21430, 21996, 95774, 49363, 12648, 54308, 6737, 8745, 11121, 49627

Offset: 1

Jeppe Stig Nielsen, Apr 24 2022

nonn

The expression is the cyclotomic polynomial value Phi_{p^2}(b) where p=prime(n).
By definition, a(n) > 1. The occurrences of a(n)=2 correspond exactly to the terms of A156585.
Does a(n) tend to infinity (is liminf a(n) infinite)?
If it exists, a(27) > 857. - J.W.L. (Jan) Eerland, Dec 23 2022
a(65) = 1624. - Serge Batalov, Nov 17 2023

MersenneForum, Primes of the form (b^p^n-1)/(b^p^(n-1)-1) search thread

Cf. A001248, A066180, A085398, A156585.

Table[k=2;Monitor[Parallelize[While[True,If[PrimeQ[(k^(Prime[n]^2)-1)/(k^Prime[n]-1)],Break[]];k++];k],k],{n,1,10}] (* J.W.L. (Jan) Eerland, Dec 22 2022 *)

forprime(p=2,,for(b=2,+oo,if(ispseudoprime(polcyclo(p^2,b)),print1(b,", ");break())))

from sympy import isprime, prime
def a(n, startb=2):
    pn = prime(n); pn2 = pn**2; b = startb
    while not isprime((b**pn2-1)//(b**pn-1)): b += 1
    return b
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jun 21 2022

a(n) = A085398(prime(n)^2) = A085398(A001248(n)).

a(25)-a(26) from J.W.L. (Jan) Eerland, Dec 23 2022
a(27) from Michael S. Branicky, Apr 04 2023
a(28)-a(33) from Martin Hopf, Nov 10 2023
a(34)-a(60) from Ryan Propper, Nov 17 2023

cp's OEIS Frontend

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

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A245730 Primes of the form 1+2^k+2^(2*k)+...+2^((j-1)*k) for some k>0, j>0.

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A006514 Primes p such that 2^p - 1 has at most 2 prime factors.

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Crossrefs

Programs

Mathematica

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A353101 Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.

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Author

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Comments

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Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A245730 Primes of the form 1+2^k+2^(2k)+...+2^((j-1)k) for some k>0, j>0.