A057468 -id:A057468 - cp's OEIS frontend

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

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

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

A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

Original entry on oeis.org

2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099

Offset: 1

Mike Oakes, Feb 23 2001

Some of the larger entries may only correspond to probable primes.

In general, for any positive integers n, a and b, a>b, a necessary condition for a^n-b^n to be prime is that either a-b=1 and n be a prime or n=1 and a-b be prime (from Arturo Magidin and Hagman in Sci.Math, Sep 11, 2010). - Vincenzo Librandi, Sep 12 2010

Cf. A000043, A057468, A059801, A059802, A062572-A062574, A062576-A062666.

is(n)=ispseudoprime(9^n-8^n) \\ Charles R Greathouse IV, Jun 13 2017

Three more terms found by Jean-Louis Charton in 2004-2005: a(9) = 30773, a(10) = 53857, a(11) = 170099. - Alexander Adamchuk, Dec 08 2006

A210506 Numbers k such that (11^k - 2^k)/9 is prime.

Original entry on oeis.org

2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421

Offset: 1

Robert Price, Jan 25 2013

All terms are prime.

a(13) > 10^5.

Cf. A004063, A028491, A057468, A059801, A121877, A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

Select[ Prime[ Range[1, 100000] ], PrimeQ[ (11^# - 2^#)/9 ]& ]

is(n)=isprime((11^n-2^n)/9) \\ Charles R Greathouse IV, Feb 17 2017

A058013 Smallest prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2

Offset: 1

Robert G. Wilson v, Nov 13 2000

The terms a(47) and a(60) [were] unknown. The sequence continues at a(48): 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, a(60)=?, continued at a(61): 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2. - Hugo Pfoertner, Aug 27 2004
In September and November 2005, Jean-Louis Charton found a(60)=54517 and a(47)=58543, respectively. Earlier, Mike Oakes found a(106)=7639 and a(124)=5839. All these large values of a(n) yield probable primes. - T. D. Noe, Dec 05 2005, Sep 18 2008
a(106) = 6529 and a(124) = 5167 are true.
a(137) is probably 196873 from prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) is 2,>32401,2,2,3,8839,5,7,2,3,5,271,13. - Robert Price, Feb 17 2012
a(276)=88301, a(139)>240000 and a(256)>100000. - Jean-Louis Charton, Jun 27 2012
Three more terms found, a(325)=81943, a(392)=64747, a(412)=56963 and also a(139)>260000, a(295)>100000, a(370)>100000, a(373)>100000. 29 unknown terms < 1000 remain. - Jean-Louis Charton, Aug 15 2012
Three more terms a(577)=55117, a(588)=60089 and a(756)=96487. - Jean-Louis Charton, Dec 13 2012
Three more (PRP) terms a(845)=83761, a(897)=48311, a(918)=54919. - Jean-Louis Charton, Dec 31 2013.
Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

Robert Price and Robert G. Wilson v, Table of n, a(n) for n = 1..138
Jean-Louis Charton and Robert G. Wilson v, a(n) for n=1..1000 status
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A065913, A103794, A115596, A247244.

Cf. A000043, A057468, A059801, A059802, A062572-A062666, A215538.

lmt = 10000; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[(n+1)^p - n^p], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[ Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)
spp[n_]:=Module[{p=2},While[!PrimeQ[(n+1)^p-n^p],p=NextPrime[p]];p]; Array[spp,90] (* Harvey P. Dale, Jul 01 2025 *)

a(n)=forprime(p=2,default(primelimit),if(ispseudoprime((n+1)^p-n^p),return(p))) \\ Charles R Greathouse IV, Feb 20 2012

a((p-1)/2) = 2 for odd primes p. - Alexander Adamchuk, Dec 01 2006

More terms from T. D. Noe, Dec 05 2005

Typo in first Mathematica program corrected by Ray Chandler, Feb 22 2017

cp's OEIS Frontend

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

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A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

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