A014232 -id:A014232 - cp's OEIS frontend

A144231 Prime numbers of the form 3^k +- 2 for k >= 1.

5, 7, 11, 29, 79, 83, 241, 727, 6563, 19681, 59051, 4782971, 14348909, 31381059607, 282429536483, 2541865828331, 150094635296999123, 450283905890997361, 36472996377170786401, 1144561273430837494885949696429

Offset: 1

Reikku Kulon, Sep 15 2008

nonn

a(49) = 3^2224 - 2 and a(50) = 3^2521 - 2 are too big for a b-file. - Robert Israel, Nov 22 2015

Robert Israel, Table of n, a(n) for n = 1..48

Cf. A000040, A000668, A014232, A057735.

A:= NULL:
for k from 1 to 1000 do
  t:= 3^k;
  if isprime(t-2) then A:= A, t-2 fi;
  if isprime(t+2) then A:= A, t+2 fi;
od:
A; # Robert Israel, Nov 22 2015

Reap[For[k = 1, k <= 100, k++, p = 3^k-2; If[PrimeQ[p], Sow[p]]; If[PrimeQ[p+4], Sow[p+4]]]][[2, 1]] (* Jean-François Alcover, Dec 18 2013 *)

A164783 a(n) = 7^n-6.

Original entry on oeis.org

1, 43, 337, 2395, 16801, 117643, 823537, 5764795, 40353601, 282475243, 1977326737, 13841287195, 96889010401, 678223072843, 4747561509937, 33232930569595, 232630513987201, 1628413597910443, 11398895185373137

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m (n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m (n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers.

Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem. Vol. 4, No. 2, Dec 1978, pp. 277-302.
Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
D. Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000420.

Cf. A014232, A014224, A135535, A059266.

[7^n-6: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 35 x)/((1-x) (1-7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[7 # + 36 & , 1, 18] (* Bruno Berselli, Feb 06 2013 *)
LinearRecurrence[{8,-7},{1,43},30] (* Harvey P. Dale, Nov 27 2014 *)

a(n)=7^n-6 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*a(n-1)+36 with n>1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
G.f.: x*(1+35*x)/((1-x)*(1-7*x)). - Colin Barker, Mar 08 2012
a(n) = 8*a(n-1) - 7*a(n-2) for n>2, a(1)=1, a(2)=43. - Vincenzo Librandi, Feb 06 2013
a(n) = A000420(n) - 6 for n>0. - Michel Marcus, Aug 31 2013

More terms a(8)-a(19) from Vincenzo Librandi, Oct 29 2009

A080892 Numbers k such that 3^k-2 is a semiprime.

Original entry on oeis.org

3, 8, 10, 12, 13, 15, 16, 19, 20, 21, 25, 28, 39, 42, 44, 48, 55, 57, 60, 66, 67, 76, 78, 85, 118, 130, 156, 162, 193, 212, 214, 217, 218, 228, 244, 312, 330, 352, 357, 376, 386, 388, 412, 442, 449, 464, 480, 525, 545, 552, 630, 644

Offset: 1

Hugo Pfoertner, Mar 30 2003

The next roadblock to being able to extend the sequence is 3^658 - 2, a 314-decimal digit composite with no known factors. - Ryan Propper, Feb 07 2013

			a(1) = 3 because 3^3-2 = 25 = 5*5.
a(2) = 8 because 3^8-2 = 6559 = 7*937.
a(3) = 10 because 3^10-2 = 59047 = 137*431.

Herman Jamke and others, Illustration of first 42 terms
Ryan Propper, Table of factorizations of 3^n - 2.
Factordb.com, Status of 3^658-2.

Cf. A001358, A014224, A014232, A080798.

Do[f = 3^n - 2; If[ !PrimeQ[f], s = FactorIntegerECM[f]; If[PrimeQ[s] && PrimeQ[f/s], Print[n]]], {n, 2, 10^3}] (* Ryan Propper, May 11 2007 *)

for(n=1,200,if(bigomega(3^n-2)==2,print1(n","))) /* Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 02 2007 */

Added missing a(1)=3 by Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 01 2007
a(27)-a(42) from Herman Jamke (hermanjamke(AT)fastmail.fm) and Ryan Propper, Apr 01, Apr 19 2007, May 11 2007
Restored missing terms < 388 by Sean A. Irvine, Apr 06 2011 (Some correctly stated terms in Jamke's and Propper's list had been omitted during editing)
a(43)-a(47) from Sean A. Irvine, Jun 13 2012
a(48) from Ryan Propper, Sep 30 2012
a(49)-a(52) from Ryan Propper, Feb 07 2013

A234503 Number of ways to write n = k + m with k > 0 and m > 0 such that 3^(phi(k)/2 + phi(m)/12) + 2 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 4, 4, 2, 3, 2, 1, 3, 4, 8, 3, 4, 4, 4, 6, 3, 4, 6, 3, 5, 5, 3, 2, 2, 6, 5, 3, 2, 3, 7, 4, 3, 4, 4, 3, 4, 4, 4, 5, 2, 5, 2, 6, 5, 7, 3, 5, 7, 6, 13, 5, 7, 7, 10, 6, 8, 8, 9, 6, 7, 8, 6, 6, 5, 7, 9, 6, 7, 8, 10

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

It might seem that a(n) > 0 for all n > 14, but a(43905) = 0. If a(n) > 0 infinitely often, then there are infinitely many primes of the form 3^m + 2.
Similarly, it might seem that for n > 26 there is a positive integer k < n such that m = phi(k)/2 + phi(n-k)/12 is an integer with 3^m - 2 prime, but n = 41213 is a counterexample.
See also A234451 and A236358 for similar sequences.

			a(15) = 1 since 15 = 1 + 14 with 3^(phi(1)/2 + phi(14)/12) + 2 = 3 + 2 = 5 prime.
a(23) = 1 since 23 = 10 + 13 with 3^(phi(10)/2 + phi(13)/12) + 2 = 3^3 + 2 = 29 prime.
a(24) = 1 since 24 = 3 + 21 with 3^(phi(3)/2 + phi(21)/12) + 2 = 3^2 + 2 = 11 prime.
a(37) = 1 since 37 = 9 + 28 with 3^(phi(9)/2 + phi(28)/12) + 2 = 3^4 + 2 = 83 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Cf. A000010, A000040, A000244, A014232, A057735, A079363, A111974, A234344, A234346, A234347, A234361, A234451, A234470, A234475, A234504, A236358.

f[n_,k_]:=3^(EulerPhi[k]/2+EulerPhi[n-k]/12)+2
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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

A164784 a(n) = 6^n-5.

Original entry on oeis.org

1, 31, 211, 1291, 7771, 46651, 279931, 1679611, 10077691, 60466171, 362797051, 2176782331, 13060694011, 78364164091, 470184984571, 2821109907451, 16926659444731, 101559956668411, 609359740010491, 3656158440062971

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m (n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m (n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (7, -6).

[6^n-5: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 24 x)/(1 - 7 x + 6 x^2), {x, 0, 30}],x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 6*a(n-1)+25 with n>1, a(1)=1. - Vincenzo Librandi, Oct 29 2009
G.f.: x*(1 + 24*x)/(1 - 7*x + 6*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 4 + (exp(5*x) - 5)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

A164785 a(n) = 5^n - 4.

Original entry on oeis.org

1, 21, 121, 621, 3121, 15621, 78121, 390621, 1953121, 9765621, 48828121, 244140621, 1220703121, 6103515621, 30517578121, 152587890621, 762939453121, 3814697265621, 19073486328121, 95367431640621, 476837158203121

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A059613.

[5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

5^Range[50]-4 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* Harvey P. Dale, Jun 07 2012 *)
CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - Harvey P. Dale, Jun 07 2012
G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009

A164786 a(n) = 8^n-7.

Original entry on oeis.org

1, 57, 505, 4089, 32761, 262137, 2097145, 16777209, 134217721, 1073741817, 8589934585, 68719476729, 549755813881, 4398046511097, 35184372088825, 281474976710649, 2251799813685241, 18014398509481977, 144115188075855865

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (9, -8).

[8^n-7: n in [1..20]]; // Vincenzo Librandi, Aug 22 2011

8^Range[20]-7 (* or *) LinearRecurrence[{9,-8},{1,57},20] (* Harvey P. Dale, Jan 24 2013 *)

a(n) = 8*a(n-1)+49, with a(1)=1. - Vincenzo Librandi, Nov 30 2010
G.f.: x*(1+48*x)/(1-9*x+8*x^2). a(n) = 9*a(n-1)-8*a(n-2). - Colin Barker, Jan 28 2012
E.g.f.: 6 + (exp(7*x) - 7)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(7)-a(19) from Vincenzo Librandi, Oct 29 2009

A093612 Primes of form 7^n-2.

Original entry on oeis.org

5, 47, 2399, 823541, 5764799, 13841287199, 4747561509941, 459986536544739960976799, 157775382034845806615042741, 97327453648743672783790144527749033795901408624680013074608083129650399

Offset: 1

Arnaud Vernier, May 23 2004

nonn

The exponents n are listed in A090669, cf. formula. [From M. F. Hasler, Nov 26 2009]

The next term (a(11)) has 83 digits. - Harvey P. Dale, Nov 14 2014

Cf. A014232.

Select[7^Range[90]-2,PrimeQ] (* Harvey P. Dale, Nov 14 2014 *)

a(n)=7^A090669(n)-2. [From M. F. Hasler, Nov 26 2009]

Terms beyond a(6) from M. F. Hasler, Nov 26 2009

A022603 Indices of primes of form 3^p - 2.

Original entry on oeis.org

4, 22, 53, 129, 2227, 1357200840

Offset: 1

Neil Fernandez

nonn

			The second prime that is 2 less than a power of 3 is 79, which is p(22), so 22 is second on the list.

Exploring Primeness Project

Cf. A014232.

cp's OEIS Frontend

A144231 Prime numbers of the form 3^k +- 2 for k >= 1.

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A164783 a(n) = 7^n-6.

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A080892 Numbers k such that 3^k-2 is a semiprime.

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A234503 Number of ways to write n = k + m with k > 0 and m > 0 such that 3^(phi(k)/2 + phi(m)/12) + 2 is prime, where phi(.) is Euler's totient function.

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A080798 Largest prime factor of 3^n-2.

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Magma

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A164784 a(n) = 6^n-5.

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Magma

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A164785 a(n) = 5^n - 4.

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