A128025 -id:A128025 - cp's OEIS frontend

A128027 Numbers n such that (11^n - 3^n)/8 is prime.

3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms < 10^5.

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, A128028, A128029, A128030, A128031, A128032.

[p: p in PrimesUpTo(400) | IsPrime((11^p-3^p) div 8)]; // Vincenzo Librandi, Nov 20 2014

A128027:=n->`if`(isprime((11^n-3^n)/8),n,NULL): seq(A128027(n),n=1..1000); # Wesley Ivan Hurt, Nov 19 2014

k=8; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]
Do[If[PrimeQ[(11^n - 3^n)/8], Print[n]], {n, 10^4}] (* Ryan Propper, Mar 17 2007 *)
Select[Prime[Range[1200]], PrimeQ[(11^# - 3^#)/8] &] (* Vincenzo Librandi, Nov 20 2014 *)

is(n)=ispseudoprime((11^n - 3^n)/8) \\ Charles R Greathouse IV, Feb 17 2017

a(8) from Ryan Propper, Mar 17 2007
a(9) from Farideh Firoozbakht, Apr 04 2007
a(10)=13103, a(11)=90397 from Robert Price, Apr 24 2011

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

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

A128066 Numbers k such that (3^k + 4^k)/7 is prime.

Original entry on oeis.org

3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.

Henri Lifchitz & Renaud Lifchitz, Top probable primes of the form (4^p+3^p)/7

Cf. A007658 = n such that (3^n + 1)/4 is prime; A057469 ((3^n + 2^n)/5); A122853 ((3^n + 5^n)/8).
Cf. A128067, A128068, A128069, A128070, A128071, A128072, A128073, A128074, A128075.
Cf. A059801 (4^n - 3^n); A121877 ((5^n - 3^n)/2).
Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

a:=proc(n) if type((3^n+4^n)/7,integer)=true and isprime((3^n+4^n)/7)=true then n else fi end: seq(a(n),n=1..1500); # Emeric Deutsch, Feb 17 2007

Do[ p=Prime[n]; f=(3^p+4^p)/(4+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

f(n)=(3^n + 4^n)/7;
forprime(n=3,10^5,if(ispseudoprime(f(n)),print1(n,", ")))
/* Joerg Arndt, Mar 27 2011 */

3 more terms from Emeric Deutsch, Feb 17 2007
2 more terms from Farideh Firoozbakht, Apr 16 2007
Two more terms (13463 and 23929) found by Lelio R Paula in 2008 corresponding to probable primes with 8105 and 14406 digits. Jean-Louis Charton, Oct 06 2010
Two more terms (81223 and 121271) found by Jean-Louis Charton in March 2011 corresponding to probable primes with 48901 and 73012 digits

A128071 Numbers k such that (3^k + 13^k)/16 is prime.

Original entry on oeis.org

3, 7, 127, 2467, 3121, 34313

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.
a(4) is certified prime by primo; a(5) is a probable prime. - Ray G. Opao, Aug 02 2007
a(7) > 10^5. - Robert Price, Apr 14 2013

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128067, A128068, A128069, A128070, A128072, A128073, A128074, A128075. 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, A128031, A128032.

k=13; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

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

One more term from Ray G. Opao, Aug 02 2007

a(6) from Robert Price, Apr 14 2013

cp's OEIS Frontend

A128027 Numbers n such that (11^n - 3^n)/8 is prime.

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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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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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