A080892 -id:A080892 - cp's OEIS frontend

A028491 Numbers k such that (3^k - 1)/2 is prime.

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117

Offset: 1

N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)

If k is in the sequence and m=3^(k-1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m))), so 3^(A028491-1) is a subsequence of A033632. For example since 9551 is in the sequence, phi(sigma(3^9550)) = sigma(phi(3^9550)). - Farideh Firoozbakht, Feb 09 2005
Salas lists these, except 3, in "Open Problems" p. 6 [March 2012], and proves that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).
Also, k such that 3^k-1 is a semiprime - see also A080892. - M. F. Hasler, Mar 19 2013

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See p. 81.
Paul Bourdelais, A Generalized Repunit Conjecture, Posting in NMBRTHRY@LISTSERV.NODAK.EDU, Jun 25, 2009.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969 [math.NT], 2012.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A076481, A033632, A112646.

Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* Farideh Firoozbakht, Feb 09 2005 *)

forprime(p=2,1e5,if(ispseudoprime(3^p\2),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(13) from Farideh Firoozbakht, Mar 27 2005
a(14)-a(16) from Robert G. Wilson v, Apr 11 2005
All larger terms only correspond to probable primes.
a(17) from Paul Bourdelais, Feb 08 2010
a(18) from Paul Bourdelais, Jul 06 2010
a(19) from Paul Bourdelais, Feb 05 2019
a(20) and a(21) from Ryan Propper, Dec 29 2021
a(22) from Ryan Propper, Nov 06 2023
a(23) from Ryan Propper, Nov 09 2023

A165767 Numbers m such that 2^m-m is a semiprime.

Original entry on oeis.org

6, 7, 15, 18, 25, 31, 33, 39, 42, 45, 49, 62, 73, 85, 93, 103, 119, 171, 187, 193, 199, 201, 269, 367, 379, 405, 413, 449, 459, 481, 489, 549, 577, 601, 631, 669, 787, 795, 1399

Offset: 1

M. F. Hasler, Oct 08 2009, Oct 29 2009

The largest resp. smallest prime factor of 2^a(n)-a(n) is listed in A165768 resp. A165769.
a(40) >= 1489. - Max Alekseyev, Aug 05 2019
1501, 1587, 1667, 2250, 3393, 5845, 9967, 16147 are terms of this sequence. - Chai Wah Wu, Oct 18 2019

			199 is in this sequence because 2^199-199 = 17377902756647509 * 46235097144973199564251065756966919577339221 and these two factors are prime.

Cf. A080892, A081715

Select[Range[1000], PrimeOmega[2^# - #]==2 &] (* Vincenzo Librandi, Dec 19 2014 *)

for( i=1,200, bigomega(2^i-i)==2 & print1(i","))

2^a(n)-a(n) = A165768(n)*A165769(n) is a semiprime.

a(n)=2k <=> 4^k/2-k is prime <=> A165768(n)=2.

More terms from Sean A. Irvine, Oct 22 2009
a(36)-a(37) from Max Alekseyev, Jun 06 2013
a(38) from Sean A. Irvine, Mar 17 2015
a(39) from Sean A. Irvine, Jun 29 2015

A080798 Largest prime factor of 3^n-2.

Original entry on oeis.org

7, 5, 79, 241, 727, 23, 937, 19681, 431, 499, 4703, 8093, 40193, 2869781, 483671, 94747, 4657, 232452293, 498112057, 2812679, 31381059607, 3765727153, 1364071, 44594137339, 125231, 13170403, 5353801183, 4159349, 46050353857, 294487079, 26892769, 29178816413, 3533781113

Offset: 2

Hugo Pfoertner, Mar 25 2003

nonn

Hugo Pfoertner, Table of n, a(n) for n = 2..210

Cf. A014224, A014232, A080443, A080892.

Cf. A006530, A058481.

[Max(PrimeDivisors(3^n-2)):n in [2..30]]; // Marius A. Burtea, Jul 12 2019

FactorInteger[#][[-1,1]]&/@(3^Range[2,30]-2) (* Harvey P. Dale, Apr 07 2022 *)

a(n) = vecmax(factor(3^n-2)[,1]); \\ Michel Marcus, Jul 12 2019

a(n) = A006530(A058481(n)). - Michel Marcus, Jul 12 2019

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

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A028491 Numbers k such that (3^k - 1)/2 is prime.

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A165767 Numbers m such that 2^m-m is a semiprime.

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A080798 Largest prime factor of 3^n-2.

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

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