A057735 -id:A057735 - cp's OEIS frontend

A228030 Primes of the form 7^n + 6.

7, 13, 349, 33232930569607, 2651730845859653471779023381607

Offset: 1

Vincenzo Librandi, Aug 11 2013

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

Cf. A217130 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=7, h=6), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  7^n+6];

Select[Table[7^n + 6, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A228031 Primes of the form 7^n + 10.

Original entry on oeis.org

11, 17, 59, 353, 2411, 117659, 823553, 1977326753, 9387480337647754305659, 3219905755813179726837617, 44567640326363195900190045974568017, 616873509628062366290756156815389726793178417, 30226801971775055948247051683954096612865741953

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..21

Cf. A217132 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=7, h=10), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  7^n+10];

Select[Table[7^n + 10, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

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

A153280 Eigensequence of triangle A153279.

Original entry on oeis.org

1, 3, 15, 165, 4785, 397155, 97302975, 71128474725, 155700231173025, 1021860617188563075, 20115326249356864131375, 1187830130350772183821825125, 210422919761508941591852499068625, 111827787746815596446398867662527275875

Offset: 0

Gary W. Adamson, Dec 23 2008

nonn

			Triangle M =
1;
1;
2, 1;
4, 2, 3;
8, 4, 6, 9;
16, 8, 12, 18, 27;
...
M^n rapidly converges to this sequence with sufficiently large n.
a(0) = 1, a(1) = 1*(2+3^0) = 3, a(2) = 3*(2+3^1) = 15, a(3) = 15*(2+3^2) = 165, a(4) = 165*(2+3^3) = 4785, ... - _Philippe Deléham_, Sep 27 2014

Harvey P. Dale, Table of n, a(n) for n = 0..65

Cf. A153279, A057735.

RecurrenceTable[{a[n+1] == a[n]*(2 + 3^n), a[0] == 1}, a, {n, 0, 15}] (* Vaclav Kotesovec, Jan 22 2023 *)
Table[2^n * QPochhammer[-1/2, 3, n], {n, 0, 15}] (* Vaclav Kotesovec, Jan 22 2023 *)
nxt[{n_,a_}]:={n+1,a(2+3^n)}; NestList[nxt,{0,1},20][[;;,2]] (* Harvey P. Dale, Mar 28 2024 *)

Given triangle A153279, let a new triangle = M shifted down one row, inserting a "1" in (0,0). Triangle equals lim_{n->oo} M^n.
a(n+1) = a(n)*(2+3^n), a(0) = 1. - Philippe Deléham, Sep 27 2014
a(n) ~ c * 3^(n*(n-1)/2), where c = QPochhammer(-2, 1/3) = 6.80914656805984199... - Vaclav Kotesovec, Jan 22 2023

More terms from Philippe Deléham, Sep 27 2014

A228027 Primes of the form 4^k + 9.

Original entry on oeis.org

13, 73, 1033, 262153, 1073741833, 73786976294838206473, 4835703278458516698824713

Offset: 1

Vincenzo Librandi, Aug 11 2013

Subsequence of A104070. - Elmo R. Oliveira, Nov 28 2023

			262153 is a term because 4^9 + 9 = 262153 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..12

Cf. A000040, A217350 (corresponding k's).

Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), A228026 (r=4, h=3), this sequence (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is 4^n+9];

Select[Table[4^n + 9, {n, 0, 200}],PrimeQ]

a(n) = 4^A217350(n) + 9. - Elmo R. Oliveira, Nov 28 2023

Corrected cross-references - Robert Price, Aug 01 2017

A228033 Primes of the form 8^k + 5.

Original entry on oeis.org

13, 2787593149816327892691964784081045188247557, 15177100720513508366558296147058741458143803430094840009779784451085189728165691397

Offset: 1

Vincenzo Librandi, Aug 11 2013

a(4) = 8^64655 + 5 = 1.919...*10^58389 is too large to include. - Amiram Eldar, Jul 23 2025

Cf. A217355 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [1..300] | IsPrime(a) where a is 8^n+5];

Select[Table[8^n + 5, {n, 4000}], PrimeQ]

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

A228028 Primes of the form 5^n + 4.

Original entry on oeis.org

5, 29, 15629, 9765629

Offset: 1

Vincenzo Librandi, Aug 11 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..8

Cf. A124621 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A228027 (k=4, h=9), A182330 (k=5, h=2), this sequence (k=5, h=4), A228029 (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A228030 (k=7, h=6), A228031 (k=7, h=10), A228032 (k=8, h=3), A228033 (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+4];

Select[Table[5^n + 4, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A267945 Primes that are a prime power plus two.

Original entry on oeis.org

5, 7, 11, 13, 19, 29, 31, 43, 61, 73, 83, 103, 109, 127, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883

Offset: 1

Robert C. Lyons, Jan 22 2016

nonn

The term 'prime power' refers to the elements of A246655.
If we were to extend the definition of prime power to include 1, then 3 would be the first term of the sequence, because 3 = 2^0 + 2.
The sequence is probably infinite, since it includes all the terms of A006512 (Greater of twin primes).
From Robert Israel, Jan 22 2016: (Start)
Since 3 divides p or p^k+2 if k is even, the only terms of the form p^k+2 where k is even are A228034.
All terms not in A057735 are congruent to 1 mod 3.
The generalized Bunyakovsky conjecture implies that for any odd k, there are infinitely many terms of the form p^k+2. (End)

			5 is in the sequence because 5 = 3^1 + 2.
7 is in the sequence because 7 = 5^1 + 2.
11 is in the sequence because 11 = 3^2 + 2.
13 is in the sequence because 13 = 11^1 + 2.
29 is in the sequence because 29 = 3^3 + 2.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Generalized Bunyakovsky conjecture

Cf. A000961, A057735, A228034, A246655, A267944.

select(t -> isprime(t) and nops(numtheory:-factorset(t-2))=1, [ seq(i,i=3..1000, 2)]); # Robert Israel, Jan 22 2016

A267945Q = PrimeQ@# && (Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2]) & (* JungHwan Min, Jan 25 2016 *)
Select[Array[Prime, 100], Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2] &] (* JungHwan Min, Jan 25 2016 *)

lista(nn) = {forprime(p=2, nn, if (isprimepower(p-2), print1(p, ", ")););} \\ Michel Marcus, Jan 22 2016

filter( is_prime, [ n+2 for n in prime_powers( 1, 1000 ) ] )

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 *)

cp's OEIS Frontend

A228030 Primes of the form 7^n + 6.

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A228031 Primes of the form 7^n + 10.

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A080443 Largest prime factor of 3^n+2.

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A153280 Eigensequence of triangle A153279.

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A228027 Primes of the form 4^k + 9.

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A228033 Primes of the form 8^k + 5.

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

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A228028 Primes of the form 5^n + 4.

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