A062666 -id:A062666 - cp's OEIS frontend

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

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

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

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

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

A062577 Numbers k such that 11^k - 10^k is prime.

Original entry on oeis.org

3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081

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 11^p-10^p

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

is(n)=ispseudoprime(11^n-10^n) \\ Charles R Greathouse IV, Feb 17 2017

Two more terms 18199 and 35153 from Jean-Louis Charton, Sep 02 2009
New term 206081 found by Jean-Louis Charton in October 2011
Edited by M. F. Hasler, Sep 16 2013

A062576 Numbers k such that 10^k - 9^k is prime.

Original entry on oeis.org

2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 1000 are often only strong pseudoprimes.

All terms are prime. - Alexander Adamchuk, Apr 27 2008

			10^2 - 9^2 = 100 - 81 = 19, which is prime, hence 2 is in the sequence.
10^3 - 9^3 = 1000 - 729 = 271, which is prime, hence 3 is in the sequence.
10^4 - 9^4 = 10000 - 6561 = 3439 = 19 * 181, which is not prime, hence 4 is not in the sequence.

Henri & Renaud Lifchitz, PRP Records.

Cf. A000043, A057468, A059801, A059802, A059803 (9^n-8^n is prime), A062572-A062666.

Cf. A016189 = 10^n - 9^n, and A199819 (primes of this form).

Select[Range[1000], PrimeQ[10^# - 9^#] &] (* Alonso del Arte, Sep 06 2013 *)

is(n)=ispseudoprime(10^n-9^n) \\ Charles R Greathouse IV, Feb 20 2017

Three more terms 15787, 66949 and 282493 found by Jean-Louis Charton in 2004 and 2007

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3

Offset: 1

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.
a(96) = 1307, a(98) = 709.
a(137) is probably 196873 from a 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) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012
a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Robert Price and Ray Chandler, Table of n, a(n) for n = 1..138 (first 136 terms from Robert Price)
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
Cf. A065913 (smallest prime of form (n+1)^k - n^k).
Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
Cf. A047845, A014076.
Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
Cf. A000043, A057468, A059801, A059802, A062572-A062666.

A062573 Numbers k such that 7^k - 6^k is prime.

Original entry on oeis.org

2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 often correspond only to strong probable primes.

			7^2 - 6^2 = 49 - 36 = 13, which is prime, so 2 is in the sequence.
7^3 - 6^3 = 343 - 216 = 127, which is prime, so 3 is in the sequence.

Henri & Renaud Lifchitz, Top probable primes of the form 7^p-6^p

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

Select[Range[100], PrimeQ[7^# - 6^#] &] (* Alonso del Arte, Sep 04 2013 *)

is(n)=ispseudoprime(7^n-6^n) \\ Charles R Greathouse IV, Feb 20 2017

Two more terms (69371 and 86689) found by Predrag Minovic in 2004 corresponding to probable primes with 58626 and 73261 digits. - Jean-Louis Charton, Oct 06 2010

New term 355039 found by Jean-Louis Charton in May 2011 corresponding to a probable prime with 300043 digits.

A062574 Numbers k such that 8^k - 7^k is prime or a strong pseudoprime.

Original entry on oeis.org

7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

All terms are prime. - Alexander Adamchuk, Apr 27 2008

Henri & Renaud Lifchitz, PRP Records.

Cf. A000043, A057468, A059801, A059802, A059803 (9^n-8^n is prime), A062572-A062666.

Cf. A016177 = 8^n - 7^n.

Select[Range[200], PrimeQ[8^# - 7^#] &] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)

is(n)=ispseudoprime(8^n-7^n) \\ Charles R Greathouse IV, Sep 04 2013

Two more terms 44029 and 76213 found by Ananda Tallur & Jean-Louis Charton in 2003.
Three more terms 83663, 173687 and 336419 found by Jean-Louis Charton in 2004 and 2008
New term 615997 found by Jean-Louis Charton corresponding to a probable prime with 556301 digits. Jean-Louis Charton, Sep 02 2009

cp's OEIS Frontend

A062572 Numbers k such that 6^k - 5^k 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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A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

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

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A062577 Numbers k such that 11^k - 10^k is prime.

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A062576 Numbers k such that 10^k - 9^k is prime.

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A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

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A062573 Numbers k such that 7^k - 6^k is prime.