A051783 -id:A051783 - cp's OEIS frontend

A138066 Least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 11, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 113, 0, 1, 7, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 12, 1, 0, 1, 3, 0, 1, 255, 0, 8, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 2, 15, 0, 2, 1, 0, 1, 23, 0, 1, 1, 0, 4, 3, 0, 1, 1, 0, 3, 1, 0, 136, 1, 0, 1

Offset: 1

Alexander Adamchuk, Mar 02 2008

a(3n+1) = 0 for n > 0.

a(84) > 100000. - Ray Chandler, Aug 10 2011

Cf. A084713 (smallest prime of the form (2n-1)^k + 2, or 0 if no such number exists).
Cf. A138067 (least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists).
Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A138051, A087886, A113481.

A080443 Largest prime factor of 3^n+2.

Original entry on oeis.org

3, 5, 11, 29, 83, 7, 43, 199, 6563, 127, 59051, 25307, 48313, 63773, 4782971, 14348909, 119243, 335429, 23203, 10613, 60089, 1224149, 795323, 919, 282429536483, 1583717027, 2541865828331, 693236134999, 174632003473

Offset: 0

Hugo Pfoertner, Mar 21 2003

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..210 (first 100 terms from Harvey P. Dale)

Cf. A051783, A057735, A080798, A081715.

[Max(PrimeDivisors(3^n+2)):n in [0..28]]; // Marius A. Burtea, Jul 12 2019

Table[FactorInteger[3^n+2][[-1,1]],{n,0,30}] (* Harvey P. Dale, Oct 21 2011 *)

for(n=0,28,f=factor(3^n+2);print1(f[#f[,1],1],", ")) \\ Hugo Pfoertner, Jul 12 2019

Corrected by T. D. Noe, Nov 15 2006

A057738 Primes p such that 3^p + 2 is prime.

Original entry on oeis.org

2, 3, 139

Offset: 1

G. L. Honaker, Jr., Oct 29 2000

Primes in A051783. - Jens Kruse Andersen, Jun 29 2014

No further terms < 1753089 using A051783. - Michael S. Branicky, May 14 2025

			a(2) = 3 because 3^3 + 2 = 29 is prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 139, p. 48, Ellipses, Paris 2008.

Cf. A051783, A057739.

[p: p in PrimesUpTo(2000) | IsPrime(3^p+2)]; // Vincenzo Librandi, Jun 30 2014

For[n = 1, n < 700, n++, If[PrimeQ[3^Prime[n] + 2], Print[Prime[n]]]] (* Stefan Steinerberger, Mar 18 2006 *)

A081715 Numbers n such that 3^n+2 is a semiprime.

Original entry on oeis.org

6, 7, 11, 12, 20, 27, 28, 40, 44, 60, 71, 84, 108, 118, 145, 156, 160, 211, 263, 295, 296, 304, 306, 316, 351, 474, 488, 495

Offset: 1

Hugo Pfoertner, Apr 04 2003

a(29) >= 514. - Hugo Pfoertner, Jul 24 2019

531, 562, 676, 760, 807, 866, 1059, 1502, 1659, 2539, 2656, 3070, 3163, 4014, 5736, 5966, 6680, 6745, 7192, 7861, 8104, 9703, 10014 are terms of this sequence. - Chai Wah Wu, Oct 18 2019

			a(1)=6 because 3^6+2=731=17*43, a(2)=7 because 3^7+2=2189=11*199.
a(1)=6 because 3^6+2=731=17*43
a(2)=7 because 3^7+2=2189=11*199
a(3)=11 because 3^11+2=177149=7*25307
a(4)=12 because 3^12+2=531443=11*48313
a(5)=20 because 3^20+2=3486784403=58027*60089
a(6)=27 because 3^27+2=7625597484989=11*693236134999
a(7)=28 because 3^28+2=22876792454963=131*174632003473
a(8)=40 because 3^40+2=12157665459056928803=1170408739*10387538177
a(9)=44 because 3^44+2=984770902183611232883=21577*45639843452917979
a(10)=60 because 3^60+2=42391158275216203514294433203=89*476305149159732623756117227
a(11)=71 because 3^71+2=7509466514979724803946715958257549=7*1072780930711389257706673708322507
a(12)=84 because 3^84+2=11972515182562019788602740026717047105683=13483993*887905769645684315365837109728331
a(13)=108 because 3^108+2=3381391913522726342930221472392241170198527451848563=671633*5034582746116891729456744192724659405059798211
a(14)=118 because 3^118+2=199667811101603467823686647723289448859052847504205678491=17*11745165358917851048452155748428791109356049853188569323
a(15)=145 because 3^145+2=1522586358169246802159262479225089070726226750574991661790882326344645=5*304517271633849360431852495845017814145245350114998332358176465268929
a(16)=156 because 3^156+2=269721605590607563262106870407286853611938890184108047911269431464974473523=21883136019044570108827*12325546272521124629737118652366725946328428459583049
a(17)=160 because 3^160+2=21847450052839212624230656502990235142567050104912751880812823948662932355203=19*1149865792254695401275297710683696586450897373942776414779622313087522755537
a(18)=211 because 3^211+2=47052721287394587764057094854672253553918218437190874778408030747195017485692977810906266281547645149=97*485079600900975131588217472728579933545548643682380152354721966465928015316422451658827487438635517
a(19)=263 because 3^263+2=304011485348815530556923313708989269910796626718253224787639751028488890841299195402970869140037716024202112537180443065484429=7*43430212192687932936703330529855609987256660959750460683948535861212698691614170771852981305719673717743158933882920437926347
a(20)=295 because 3^295+2=563339419994190847700930153835754386693266237141306322927902016783411511018514718493004963603658195013376479179415613344911575031957595780109=3535513*159337391771488564092659298335419608609349261943402929908022404891004929417177851840172830252259911083165718575894251653129708484159893

Status of 3^514+2 in factordb.com.

Cf. A001358, A051783, A057735, A080443, A080892.

for(n=1, 295, if(bigomega(3^n+2)==2, print1(n", "))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 25 2007

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 25 2007

More terms from Sean A. Irvine, Mar 21 2010

A134753 Numbers k such that 3^(2*k-1) + 2 is prime.

Original entry on oeis.org

1, 2, 8, 32, 62, 70, 118, 122, 158, 182, 196, 566, 752, 3602, 21896, 22768, 53072

Offset: 1

N. J. A. Sloane, Jan 28 2008

From Enrique Pérez Herrero, Jul 29 2010: (Start)
With: f(n)=3^(2n-1)+2, the non-primality of f(n) is settled when:
if 2 does not divide n, 5 divides f(n) (n>1)
if 3 divides n, 7 divides f(n)
if 5 divides n-4, 11 divides f(n)
if 14 divides n-2, 29 divides f(n)
if 15 divides n-5, 31 divides f(n). (End)

Luis H. Gallardo, Posting to the Number Theory List, Jan 14 2008

L. H. Gallardo, On a remark of Makowski about perfect numbers, El. Mathem. 65 (3) (2010) 121-126
Henri & Renaud Lifchitz, PRP Records.

Cf. A051783, A134752.

MaxVal = 1000;
dataA134753=Select[Select[Range[MaxVal],#<3||Mod[#,2]==0&&Mod[#,3]!=0&&Mod[#,5]!=4&&Mod[#,14]!=2&&Mod[#,15]!=5&],PrimeQ[3^(2*#-1)+2]&]
A134753[n_Integer] := dataA134753[[n]] /; (n > 0 && n <= Length[dataA134753]) (* Enrique Pérez Herrero, Jul 29 2010 *)

is(n)=isprime(3^(2*n-1)+2) \\ Charles R Greathouse IV, Jun 06 2017

({odd terms in A051783} + 1)/2.

Typo in prime search corrected Enrique Pérez Herrero, Jul 31 2010

a(15)-a(17) from A051783 by Ray Chandler, Aug 06 2011

A138067 Least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

2, 2, 3, 0, 2, 5, 0, 2, 105, 0, 2, 11, 0, 5, 3, 0, 2, 15, 0, 2, 9, 0, 2, 113, 0, 5, 7, 0, 2, 27, 0, 3, 3, 0, 3, 3, 0, 12, 61, 0, 2, 3, 0, 4, 255, 0, 8, 63, 0, 2, 9, 0, 2, 3473, 0, 2, 3, 0, 2, 15, 0, 2, 87, 0, 3, 23, 0, 36, 1861, 0, 4, 3, 0, 2, 5, 0, 3, 7, 0, 136, 425, 0, 11

Offset: 1

Alexander Adamchuk, Mar 02 2008

a(3n+1) = 0 for n > 0.

a(84) > 100000. - Ray Chandler, Aug 10 2011

Cf. A084713 (smallest prime of the form (2n-1)^k + 2, or 0 if no such number exists).
Cf. A138066 (least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists).
Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A138051, A087886, A113481.

a(54)-a(83) from Donovan Johnson, Oct 29 2008

A301919 a(n) is the least value of k for which A301918(n) divides 3^k+3.

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 15, 16, 10, 5, 22, 27, 6, 12, 7, 40, 45, 25, 51, 18, 57, 64, 69, 70, 75, 26, 40, 82, 87, 9, 99, 100, 106, 112, 117, 61, 129, 135, 16, 141, 142, 147, 18, 159, 166, 85, 88, 177, 62, 94, 190, 195, 100, 201, 103, 74, 225, 115, 231, 232, 244, 84

Offset: 1

Luke W. Richards, Mar 28 2018

nonn

This can be used to identify P+1 values to primality test potential primes P of the form 3^k+2, i.e., A051783.

			All values of 3^k+3 are multiples of 2, so 3^0+3 = 4 is the least value of k which is a multiple of 2.
a(10) = 5 and A301918(10) = 41 so 3^5+3 = 246 is the first multiple of 41 which can be written in the form 3^k+3.

Cf. A051783, A178674, A301916, A301917, A301918.

a(n) = A301917(n-1) + 1 for n > 2.

A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 17, 25, 81, 241

Offset: 1

Jianing Song, Jun 04 2018

nonn

Start of run of 3 consecutive numbers in A033948.
The next term is 3^541 - 2, which is too large to be included here. No more terms below 3^100000, or approximately 1.33*10^47712.
There is a multiple of 4 in every four consecutive positive integers and it clearly has no primitive roots if it is larger than 4. Again, there is a multiple of 3 in every three consecutive positive integers, so it must be a power of 3 or two times a power of 3, and the other two numbers must be odd prime powers or two times odd prime powers.
According to Pillai's conjecture, there're only finitely many solutions to |3^a - p^b| = 2, |3^a - 2*p^b| = 1, |p^a - 2*3^b| = 1 with a,b >= 2, p odd primes (no solution other than 3^3 - 5^2 = 2, 3^5 - 2*11^2 = 1 below 3^100000). So beyond (25, 26, 27) and (241, 242, 243), it's very likely that all three consecutive numbers with primitive roots are of the form (3^i, 3^i + 1, 3^i + 2), (3^j - 2, 3^j - 1, 3^j), (2*3^k - 1, 2*3^k, 2*3^k + 1) such that (3^i + 1)/2, 3^i + 2, 3^j - 2, (3^j - 1)/2, 2*3^k - 1, 2*3^k + 1 are primes, which only produces one more solution (3^541 - 2, 3^541 - 1, 3^541) below 3^1000000.

			81, 82, 83 all have primitive roots (in fact, their least common primitive root is 47), so 81 is a term.
Note that A014224 and A028491 have a term 541 in common, so 3^541 - 2, 3^541 - 1 and 3^541 all have primitive roots, so 3^541 - 2 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A003306, A003307, A014224, A028491, A051783, A171381, A319450.

A132829 Numbers k such that 3^k + 2 is not prime.

Original entry on oeis.org

5, 6, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

Artur Jasinski, Sep 03 2007

nonn

Cf. A057735, A051783.

[n: n in [0..100]| not IsPrime(3^n+2)]; // Vincenzo Librandi, Jan 28 2011

a = {}; c = 3^x + 2; Do[If[PrimeQ[c],0, AppendTo[a, x]], {x, 0, 100}]; a
Select[Range[90],CompositeQ[3^#+2]&] (* Harvey P. Dale, Sep 25 2021 *)

A132830 Numbers of the form 3^n+2 which are not primes.

Original entry on oeis.org

245, 731, 2189, 19685, 177149, 531443, 1594325, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 847288609445, 7625597484989, 22876792454963, 68630377364885, 205891132094651

Offset: 1

Artur Jasinski, Sep 03 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A057735, A051783, A132829.

a = {}; c = 3^x + 2; Do[If[PrimeQ[c],0, AppendTo[a, c]], {x, 0, 100}]; a (*Artur Jasinski*)
Select[3^Range[0,30]+2,!PrimeQ[#]&] (* Harvey P. Dale, Nov 21 2012 *)

cp's OEIS Frontend

A138066 Least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

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A080443 Largest prime factor of 3^n+2.

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A057738 Primes p such that 3^p + 2 is prime.

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Magma

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A081715 Numbers n such that 3^n+2 is a semiprime.

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A134753 Numbers k such that 3^(2*k-1) + 2 is prime.

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A138067 Least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

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A301919 a(n) is the least value of k for which A301918(n) divides 3^k+3.

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A305237 Numbers m such that m, m+1 and m+2 all have primitive roots.

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A132829 Numbers k such that 3^k + 2 is not prime.

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