A059802 -id:A059802 - cp's OEIS frontend

A059801 Numbers k such that 4^k - 3^k is prime.

2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233

Offset: 1

Mike Oakes, Feb 23 2001

Some of the larger entries may only correspond to probable primes.
The values corresponding to 1193 (719 digits) and 1993 (1200 digits) have been certified prime with Primo. - Rick L. Shepherd, Sep 10 2002
8 more terms found by Jean-Louis Charton during 2004 - 2006. Corresponding numbers of decimal digits are 14461, 27876, 28972, 41300, 54958, 79170, 89505, 111058, 205443. - Alexander Adamchuk, Dec 02 2006

Bogley, William A.; Williams, Gerald Efficient finite groups arising in the study of relative asphericity. Math. Z. 284, No. 1-2, 507-535 (2016).

Cf. A000043, A057468, A059802, etc., A129736, A129737.

Select[Range[1000], PrimeQ[4^# - 3^#] &] (* Alonso del Arte, Nov 03 2011 *)

select(vector(10^5,i,i),n->ispseudoprime(4^n-3^n)) \\ Charles R Greathouse IV, Feb 11 2011

A062572 Numbers k such that 6^k - 5^k is prime.

Original entry on oeis.org

2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

The 809- and 1176-digit numbers associated with the terms 1039 and 1511 have been certified prime with Primo. - Rick L. Shepherd, Nov 15 2002

			2 is in the sequence because 6^2 - 5^2 = 36 - 25 = 11, which is prime.
3 is not in the sequence because 6^3 - 5^3 = 216 - 125 = 91 = 7 * 13, which is not prime.

Cf. A000043, A057468, A059801, A059802, A062573-A062666.

Select[Range[1000], PrimeQ[6^# - 5^#] &] (* Alonso del Arte, Sep 04 2013 *)

forprime(p=2,1e4,if(ispseudoprime(6^n-5^n),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

Edited by T. D. Noe, Oct 30 2008
Two more terms (31237 and 60413) found by Predrag Minovic in 2004 corresponding to probable primes with 24308 and 47011 digits. Jean-Louis Charton, Oct 06 2010
Two more terms (113177 and 135647) found by Jean-Louis Charton in 2009 corresponding to probable primes with 88069 and 105554 digits. Jean-Louis Charton, Oct 13 2010
a(15) from Jean-Louis Charton, Apr 08 2013

A062666 Numbers k such that 100^k - 99^k is prime.

Original entry on oeis.org

2, 5, 19, 59, 1013, 2371, 13967, 44683

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 10000 correspond to probable primes.

a(9) > 10^5. - Robert Price, Jul 10 2013

Cf. A000043, A057468, A059801, A059802, A062572-A062665.

lst={}; k=100; Do[If[PrimeQ[k^n-(k-1)^n], Print[n]; AppendTo[lst, n]], {n, 10^5}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

forprime(p=2,1e5,if(ispseudoprime(100^p-99^p),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

Edited by T. D. Noe, Oct 30 2008

a(7)-a(8) from Robert Price, Jul 10 2013

A128344 Numbers k such that (7^k - 5^k)/2 is prime.

Original entry on oeis.org

3, 5, 7, 113, 397, 577, 7573, 14561, 58543, 100019, 123407, 136559, 208283, 210761, 457871, 608347, 636043

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.
No other terms less than 10^5. - Robert Price, May 28 2012
No other terms less than 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. A062572, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354. Cf. A004061, A082182, A121877, A059802. Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.

k=7; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((7^n-5^n)/2) \\ Charles R Greathouse IV, Feb 17 2017

a(7)-a(9) from Robert Price, May 28 2012

a(10)-a(17) from Jon Grantham, Jul 29 2023

A062589 Numbers k such that 23^k - 22^k is prime, or a strong pseudoprime.

Original entry on oeis.org

229, 241, 673, 5387, 47581

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 often correspond to "unproven" strong pseudoprimes.

a(6) > 10^5. - Robert Price, Aug 22 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062588, A214658, A214655, A062592-A062666.

a(5) from Robert Price, Aug 22 2012

Edited by M. F. Hasler, Sep 21 2013

A062587 Numbers k such that 21^k - 20^k is prime.

Original entry on oeis.org

2, 19, 41, 43, 337, 479, 9127, 37549, 44017, 59971, 128327, 176191, 193601

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 may correspond to (unproven) strong pseudoprimes.

Top probable primes of the form 21^p-20^p

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

is(n)=ispseudoprime(21^n-20^n) \\ Charles R Greathouse IV, Jun 13 2017

a(8) from Jean-Louis Charton, Feb 29 2012
a(9) and a(10) from Robert Price, May 30 2012
Edited by M. F. Hasler, Sep 16 2013
a(11) added by Jean-Louis Charton, Nov 24 2014
a(12) added by Jean-Louis Charton, Feb 05 2015
a(13) added by Jean-Louis Charton, Feb 18 2015

A128336 Numbers k such that (6^k + 5^k)/11 is prime.

Original entry on oeis.org

3, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

No other terms less than 100000. - Robert Price, May 11 2012

Cf. A057171, A082387, A122853, A128335, A128337, A128338, A128339, A128340, A128341, A128342, A128343, A004061, A082182, A121877, A059802, A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=6; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

forprime(p=3,1e4,if(ispseudoprime((6^p+5^p)/11),print1(p", "))) \\ Charles R Greathouse IV, Jul 16 2011

a(7)-a(9) from Alexander Adamchuk, May 04 2010

One more term (8783) added (unknown discoverer) corresponding to a probable prime with 6834 digits by Jean-Louis Charton, Oct 06 2010

A128347 Numbers k such that (11^k - 5^k)/6 is prime.

Original entry on oeis.org

5, 41, 149, 229, 263, 739, 3457, 20269, 98221

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(10) > 10^5. - Robert Price, Jan 24 2013

Cf. A062572, A128344, A128345, A128346, A128348, A128349, A128350, A128351, A128352, A128353, A128354. Cf. A004061, A082182, A121877, A059802. Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.

k=11; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((11^n-5^n)/6) \\ Charles R Greathouse IV, Feb 17 2017

a(7)-a(9) from Robert Price, Jan 24 2013

A128342 Numbers k such that (13^k + 5^k)/18 is prime.

Original entry on oeis.org

13, 19, 31, 359, 487, 757, 761, 1667, 2551, 3167, 6829

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.
No other terms below 23600. - Max Alekseyev, Feb 01 2010
a(12) > 10^5. - Robert Price, Apr 30 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128343. Cf. A004061, A082182, A121877, A059802. Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=13; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((13^n+5^n)/18) \\ Charles R Greathouse IV, Feb 17 2017

Four more terms from Max Alekseyev, Feb 01 2010

A128341 Numbers k such that (12^k + 5^k)/17 is prime.

Original entry on oeis.org

3, 5, 13, 347, 977, 1091, 4861, 4967, 34679

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(10) > 10^5. - Robert Price, May 05 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128342, A128343.
Cf. A004061, A082182, A121877, A059802.
Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=12; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]
Select[Range[1100],PrimeQ[(12^#+5^#)/17]&] (* Harvey P. Dale, Jul 24 2012 *)

is(n)=isprime((12^n+5^n)/17) \\ Charles R Greathouse IV, Feb 17 2017

Two more terms (a(7) and a(8)) from Harvey P. Dale, Jul 24 2012

a(9) from Robert Price, May 05 2013

cp's OEIS Frontend

A059801 Numbers k such that 4^k - 3^k is prime.

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A062572 Numbers k such that 6^k - 5^k is prime.

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A062666 Numbers k such that 100^k - 99^k is prime.

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A128344 Numbers k such that (7^k - 5^k)/2 is prime.

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A062589 Numbers k such that 23^k - 22^k is prime, or a strong pseudoprime.

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A062587 Numbers k such that 21^k - 20^k is prime.

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A128336 Numbers k such that (6^k + 5^k)/11 is prime.

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A128347 Numbers k such that (11^k - 5^k)/6 is prime.

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