A058765 -id:A058765 - cp's OEIS frontend

A057468 Numbers k such that 3^k - 2^k is prime.

2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503

Offset: 1

Robert G. Wilson v, Sep 09 2000

Some of the larger entries may only correspond to probable primes.
The 1137- and 1352-digit values associated with the terms 2381 and 2833 have been certified prime with Primo. - Rick L. Shepherd, Nov 12 2002
Or, numbers k such that A001047(k) is prime. - Zak Seidov, Sep 17 2006
3^k - 2^k were proved prime for k = 3613, 3853, 3929, 5297, 7417 with Primo. - David Harrison, Jun 08 2011

Henri Lifchitz and Renaud Lifchitz, PRP Records.
R. Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc. 365 (2013), 5503-5524.
Primality certificates for 3613 to 7417

Cf. A058765, A000043 (Mersenne primes), A001047 (3^n-2^n).

Subset of A000040.

Select[Prime@ Range@ 941, PrimeQ[3^# - 2^#] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 and modified by Robert G. Wilson v, Mar 15 2017 *)
ParallelMap[ If[ PrimeQ[3^# - 2^#], #, Nothing] &, Prime@ Range@ 941] (* Robert G. Wilson v, Jun 28 2017 *)

select(p->ispseudoprime(3^n-2^n), primes(100)) \\ Charles R Greathouse IV, Feb 11 2011

a(20) = 90217 found by Mike Oakes, Feb 23 2001
Terms a(21) = 122219, a(22) = 173191, a(23) = 256199 were found by Mike Oakes in 2003-2005. Corresponding numbers of decimal digits are 58314, 82634, 122238.
a(24) = 336353 found by Mike Oakes, Oct 15 2007. It corresponds to a probable prime with 160482 decimal digits.
a(25) = 485977 found by Mike Oakes, Sep 06 2009; it corresponds to a probable prime with 231870 digits. - Mike Oakes, Sep 08 2009
a(26) = 591827 found by Mike Oakes, Aug 25 2009; it corresponds to a probable prime with 282374 digits.
a(27) = 1059503 found by Mike Oakes, Apr 12 2012; it corresponds to a probable prime with 505512 digits. - Mike Oakes, Apr 14 2012

A082869 3^n - 2^n is a semiprime.

Original entry on oeis.org

4, 7, 9, 13, 19, 23, 37, 71, 89, 97, 131, 167, 193, 227, 229, 257, 263, 269, 271

Offset: 1

Hugo Pfoertner, May 24 2003

a(20) >= 653. - Max Alekseyev, Aug 26 2021

			a(1) = 4 because 3^4 - 2^4 = 5 * 13
a(2) = 7 because 3^7 - 2^7 = 29 * 71
a(3) = 9 because 3^9 - 2^9 = 1009 * 19
a(4) = 13 because 3^13 - 2^13 = 53 * 29927
a(5) = 19 because 3^19 - 2^19 = 1559 * 745181
a(6) = 23 because 3^23 - 2^23 = 47 * 2002867877
a(7) = 37 because 3^37 - 2^37 = 8891471 * 50642213021
a(8) = 71 because 3^71 - 2^71 = 67049419 * 111998979662707645844109121
a(9) = 89 because 3^89 - 2^89 = 4120081168939 * 706132008101135602203621405289
a(10) = 97 because 3^97 - 2^97 = 319128643 * 59813046375181769306016700165290169537
a(11) = 131 because 3^131 - 2^131 = 263 * 1210399177182288006201752262354382648158190136861552303421773
a(12) = 167 because 3^167 - 2^167 = 167884386911 * 284602839755962600307038183361142274453177384697761703968640951718869
a(13) = 193 because 3^193 - 2^193 = 773 * 157116815095122696291789672145814943987605497895096234870661710074857006307174092298131047
a(14) = 227 because 3^227 - 2^227 = 167360891302418779411 * 12102381564694515014432350438002672779054341887509579790377508212702751544613632122970969
a(15) = 229 because 3^229 - 2^229 = 271117470516046849 * 67237232094433305864393166477037402086197319313004074022941345112953840883539481643687544179
a(16) = 257 because 3^257 - 2^257 = 3650201327 * 114247220844165289049224917003868019618046824570124111266639206512722372880755761151052076786187552795911804402733
a(17) = 263 because 3^263 - 2^263 = 1789696394587605010251024191 * 169867630212703250249981022070263878299079238108093021871181171428200213741587995035055139427113909
a(18) = 269 because 3^269 - 2^269 = 3767 * 58833122596041019850277965408508940208380870952125838087379156948993498251689923575161076689330121444393974916753840891087813
a(19) = 271 because 3^271 - 2^271 = 2711 * 735750407736473144959046057264728365874119021724332398617327934122565857164514694088659506296666818455309890905079155116415309

P. C. Leyland, Homogeneous Cunningham numbers.
Status of 3^653-2^653 in factordb.com.

Cf. A001047, A001358, A057468, A058765, A050244, A095027.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 30 2007

A121091 Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime.

Original entry on oeis.org

7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, 5613125740675652943160572913465695837595324940170321, 371281, 919

Offset: 2

Alexander Adamchuk, Aug 11 2006, revised Dec 01 2006, Feb 15 2007

nonn

a(19) = 19^1607 - 18^1607, which is too large to include. It has 2055 decimal digits. See A062585(1) = 1607.

a(20)-a(21) = {723901, 8005616640331026125580781}. a(n) is currently known for all n up to n = 96. Corresponding smallest odd primes p such that (n+1)^p - n^p is prime are listed in A125713(n) = {3,3,3,3,5,3,7,7,3,3,3,17,3,3,43,5,3,10957,5,19,127,229,3,3,3,13,3,3,149,3,5,3,23,3,5,83,3,3,37,7,3,3,37,5,3,5,58543,...}. a(n+1) = A065013(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.

Cf. A121620, A000043, A058765, A121616, A121618.
Cf. A125713 = Smallest odd prime p such that (n+1)^p - n^p is prime. Cf. A065913 = Smallest prime of form (n+1)^k - n^k. Cf. A058013 = Smallest prime p such that (n+1)^p - n^p is 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.

a(n) = n^A125713(n) - (n-1)^A125713(n).

A161470 Primes of the form 3^k+2^k+k^3-k^2.

Original entry on oeis.org

5, 17, 53, 2609, 1604543, 7625731721669, 67585198634826967968486182915129003

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 10 2009

nonn

The associated k are 1, 2, 3, 7, 13, 27, 73, 994, 1129, ... - R. J. Mathar, Jun 12 2009

The next term has 475 digits. - Harvey P. Dale, Dec 12 2010

Cf. A058765, A082101, A161469.

[ a: n in [1..450] | IsPrime(a) where a is 3^n+2^n+n^3-n^2] // Vincenzo Librandi, Nov 30 2010

A161470 = {}; Do[If[PrimeQ[p = (3^n + 2^n) + (n^3 - n^2)], AppendTo[A161470, p]], {n, 6!}]; A161470 (* Orlovsky *)
(* Alternate: *) Select[Table[3^k + 2^k + k^3 - k^2, {k, 2000}], PrimeQ] (* Harvey P. Dale, Dec 12 2010 *)

Definition simplified by R. J. Mathar, Jun 12 2009

A095027 Semiprimes of the form 3^n - 2^n.

Original entry on oeis.org

65, 2059, 19171, 1586131, 1161737179, 94134790219, 450283768452043891, 7509466514977363620705281135650699, 2909321189362570189660446183802104997118371, 19088056323407826916968161259086927505582748291

Offset: 1

Hugo Pfoertner, Jun 03 2004

nonn

			a(1)=65 because 3^4-2^4=65=5*13 is a semiprime; a(3)=19171: 3^9-2^9=19171=19*1009.

Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A082869 = n such that 3^n-2^n is a semiprime, A058765 primes of the form 3^n-2^n.

IsSemiprime:=func; [s: n in [2..100] | IsSemiprime(s) where s is 3^n - 2^n]; // Vincenzo Librandi, Sep 21 2012

Select[Table[3^n - 2^n, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)

a(10) from Vincenzo Librandi, Sep 21 2012

A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

Original entry on oeis.org

5, 19, 13, 211, 7, 29, 71, 97, 1009, 11, 23, 331, 61, 53, 29927, 463, 3571, 17, 401, 129009091, 577, 1559, 745181, 4621, 43, 6217, 35839, 47, 2002867877, 5521, 101, 39756701, 79, 4057, 397760329, 369181, 68629840493971, 31, 241, 617671248800299, 3041, 14177

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A001047 term-by-term, factorize each term, write down any primes not seen before.

Amiram Eldar, Table of n, a(n) for n = 1..1893
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A001047, A058765, A057468; A003462, A076481, A028491, A129733; A000225, A004668, A000043, A108974.

a(41) and a(42) switched by Amiram Eldar, Jun 30 2023

A161469 Primes of the form 3^k + 2^k - k^3 + k^2.

Original entry on oeis.org

2, 5, 17, 613, 129266611, 7625731683761, 150094704016430497, 2503155504994422192936289397051173, 4638397686588101984398752568803509060305779468709

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 10 2009

nonn

The associated values of k are 0, 1,3,6,17,27,36,70,102,237,377,463,576,639,683,... - R. J. Mathar, Jun 12 2009 [Amended by Harvey P. Dale, Jan 23 2023]

The next term (a(10)) has 114 digits. - Harvey P. Dale, Jan 23 2023

Cf. A058765, A082101, A161470.

[ a: n in [1..450] | IsPrime(a) where a is 3^n+2^n-n^3+n^2]; // Vincenzo Librandi, Nov 30 2010

lst={};Do[If[PrimeQ[p=(3^n+2^n)-(n^3-n^2)],AppendTo[lst,p]],{n,0,6!}];lst
Select[Table[3^k+2^k-k^3+k^2,{k,0,200}],PrimeQ] (* Harvey P. Dale, Jan 23 2023 *)

Definition simplified by R. J. Mathar, Jun 12 2009

a(1) = 2 prepended by Harvey P. Dale, Jan 23 2023

A219283 Primes of the form 13^k - 12^k.

Original entry on oeis.org

6431804812640900941, 31211427601852046808999765129652549, 4519079836942618423019040742735616921552429101, 22137406298265966315641393147750228275603823278911109

Offset: 1

Vincenzo Librandi, Nov 23 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..7

Cf. A058765, A147667, A147669, A199820, A082101, A081505.

Cf. A062579 (associated k).

[a: n in [0..200] | IsPrime(a) where a is  13^n - 12^n];

Select[Table[13^n - 12^n, {n, 0, 200}], PrimeQ]

cp's OEIS Frontend

A057468 Numbers k such that 3^k - 2^k is prime.

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A082869 3^n - 2^n is a semiprime.

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A121091 Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime.

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A161470 Primes of the form 3^k+2^k+k^3-k^2.

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A095027 Semiprimes of the form 3^n - 2^n.

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A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

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A161469 Primes of the form 3^k + 2^k - k^3 + k^2.

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Magma

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A219283 Primes of the form 13^k - 12^k.

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