A028491 -id:A028491 - cp's OEIS frontend

A128024 Numbers k such that (7^k - 3^k)/4 is prime.

Original entry on oeis.org

3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, 187507, 353501, 817519

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes. No other terms < 1000000.

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A028491, A057468, A059801, A121877.

Cf. A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=4; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

forprime(p=3,1e5,if(ispseudoprime((7^p-3^p)/4),print1(p", "))) \\ Charles R Greathouse IV, Jun 01 2011

from sympy import isprime
def aupto(lim): return [k for k in range(lim+1) if isprime((7**k-3**k)//4)]
print(aupto(900)) # Michael S. Branicky, Mar 07 2021

a(8)-a(9) from Farideh Firoozbakht, Apr 08 2007
a(10) from Robert Price, Jun 01 2011
a(11)-a(13) from Jon Grantham, Jul 29 2023

A128026 Numbers n such that (10^n - 3^n)/7 is prime.

Original entry on oeis.org

2, 3, 5, 37, 599, 38393, 51431, 118681, 376417

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms < 1000000.

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime.
Cf. A057468 = numbers n such that 3^n - 2^n is prime.
Cf. A059801 = numbers n such that 4^n - 3^n is prime.
Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime.
Cf. A128024, A128025, A128027, A128028, A128029, A128030, A128031, A128032.

k=7; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

forprime(p=2,1e4,if(ispseudoprime((10^p-3^p)/7),print1(p", "))) \\ Charles R Greathouse IV, Jun 05 2011

a(6)-a(7) from Robert Price, Jun 04 2011

a(8)-a(9) from Jon Grantham, Jul 29 2023

A128028 Numbers k such that (13^k - 3^k)/10 is prime.

Original entry on oeis.org

7, 31, 41, 269, 283, 7333, 8803

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms exist < 100000.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128029, A128030, A128031, A128032.

k=10; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((13^n-3^n)/10) \\ Charles R Greathouse IV, Feb 17 2017

One more term from Farideh Firoozbakht, Apr 03 2007

a(7)=8803 from Robert Price, Aug 12 2011

A007658 Numbers k such that (3^k + 1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, 1896463, 2533963, 2674381, 7034611

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Prime repunits in base -3.

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).

Paul Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.

lst={};Do[If[PrimeQ[(3^n+1)/4], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime((3^n+1)/4) \\ Charles R Greathouse IV, Apr 29 2015

a(20) from Robert G. Wilson v, Apr 11 2005
a(22) from Paul Bourdelais, Nov 08 2007
a(23) from Paul Bourdelais, Apr 07 2008
a(24) from Paul Bourdelais, Apr 05 2010
a(25) from Paul Bourdelais, Aug 28 2015
a(26) from Paul Bourdelais, Jan 30 2020
a(27) from Paul Bourdelais, Mar 06 2020
a(28) from Paul Bourdelais, Mar 22 2024
a(29) from Paul Bourdelais, Dec 04 2024

A128025 Numbers k such that (8^k - 3^k)/5 is prime.

Original entry on oeis.org

2, 3, 7, 19, 31, 67, 89, 9227, 43891, 854149

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.
Verified the first 8 terms in sequence. Also, the next number in the sequence, if one exists is > 43691. - Robert Price, Mar 16 2010
a(10) > 10^5. - Robert Price, Jul 27 2011
a(11) > 10^6. - Jon Grantham, Jul 29 2023

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=5; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((8^n-3^n)/5) \\ Charles R Greathouse IV, Feb 17 2017

9227 from Farideh Firoozbakht, Apr 08 2007
a(9) from Robert Price, Jul 27 2011
a(10) from Jon Grantham, Jul 29 2023

A128031 Numbers k such that (17^k - 3^k)/14 is prime.

Original entry on oeis.org

3, 11, 17, 491, 23029

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms < 100000.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime.
Cf. A057468 = numbers n such that 3^n - 2^n is prime.
Cf. A059801 = numbers n such that 4^n - 3^n is prime.
Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime.
Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128032.

k=14; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((17^n-3^n)/14) \\ Charles R Greathouse IV, Feb 17 2017

a(5)=23029 from Robert Price, Nov 03 2011

A128032 Numbers k such that (19^k - 3^k)/16 is prime.

Original entry on oeis.org

73, 271, 421, 2711

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms <= 10^5. - Robert Price, Aug 27 2011

Cf. A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031.

k=16; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((19^n-3^n)/16) \\ Charles R Greathouse IV, Feb 17 2017

2711 from Farideh Firoozbakht, Apr 07 2007

A128029 Numbers n such that (14^n - 3^n)/11 is prime.

Original entry on oeis.org

2, 5, 13, 67, 2657, 3547, 15649

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.
There is no further term up to prime(1400)=11657. - Farideh Firoozbakht, Apr 04 2007
No other terms < 100,000.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128030, A128031, A128032.

k=11; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((14^n-3^n)/11) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Farideh Firoozbakht, Apr 04 2007

Added term a(7)=15649 by Robert Price, Sep 12 2011

A128030 Numbers k such that (16^k - 3^k)/13 is prime.

Original entry on oeis.org

2, 3, 31, 467, 1747, 29683

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms < 100000.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128031, A128032.

k=13; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]

is(n)=isprime((16^n-3^n)/13) \\ Charles R Greathouse IV, Feb 17 2017

1747 from Farideh Firoozbakht, Apr 08 2007

a(6)=29683 from Robert Price, Sep 13 2011

A001562 Numbers n such that (10^n + 1)/11 is a prime.

Original entry on oeis.org

5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787

Offset: 1

N. J. A. Sloane

The a(10) to a(11) gap represents the largest relative gap seen so far in searching repunits with bases between -12 and 12. On average, there should have been 4 more primes added to this sequence by a(11), instead of just 1. - Paul Bourdelais, Feb 11 2010

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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture, 2009.
J. Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Equals 2*A054416 + 1.

Odd terms of A309358.

Select[Range[3000], PrimeQ[(10^# + 1) / 11] &] (* Vincenzo Librandi, Oct 29 2017 *)

isok(n) = (denominator(p=(10^n+1)/11)==1) && isprime(p); \\ Michel Marcus, Oct 29 2017

a(11) corresponds to a probable prime discovered by Paul Bourdelais, Feb 11 2010

a(12) corresponds to a probable prime discovered by Paul Bourdelais, May 04 2020

cp's OEIS Frontend

A128024 Numbers k such that (7^k - 3^k)/4 is prime.

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A128026 Numbers n such that (10^n - 3^n)/7 is prime.

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A128028 Numbers k such that (13^k - 3^k)/10 is prime.

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A007658 Numbers k such that (3^k + 1)/4 is prime.

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A128025 Numbers k such that (8^k - 3^k)/5 is prime.

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A128031 Numbers k such that (17^k - 3^k)/14 is prime.

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A128032 Numbers k such that (19^k - 3^k)/16 is prime.

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A128029 Numbers n such that (14^n - 3^n)/11 is prime.

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