A082101 -id:A082101 - cp's OEIS frontend

A007689 a(n) = 2^n + 3^n.

2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 169
Index entries for linear recurrences with constant coefficients, signature (5,-6).

For odd-indexed members divided by 5 see A096951.
Binomial transform of A000051.
Cf. A000079, A000244, A001550, A063376, A063481.
Cf. A074600 - A074624, A082101 (primes).

a007689 n = a000079 n + a000244 n  -- Reinhard Zumkeller, Apr 28 2013

[2^n+3^n: n in [0..30]]; // G. C. Greubel, Mar 11 2023

A007689:=n->2^n + 3^n: seq(A007689(n), n=0..50); # Wesley Ivan Hurt, Jan 24 2017

Table[2^n + 3^n, {n, 0, 25}]
a=2;Numerator[Table[a=2*a-((a+1)/2),{n,0,7!}]] (*10 times (or more) faster for large numbers.*) (* Vladimir Joseph Stephan Orlovsky, Apr 19 2010 *)
LinearRecurrence[{5,-6},{2,5},30] (* nearly 20 times faster than the above program for large numbers. *) (* Harvey P. Dale, Oct 20 2013 *)

a(n)=2^n+3^n \\ Charles R Greathouse IV, Jun 15 2011

[lucas_number2(n,5,6)for n in range(0,27)] # Zerinvary Lajos, Jul 08 2008

E.g.f.: exp(2*x)*(1+exp(x)).
G.f.: (2-5*x)/((1-2*x)*(1-3*x)).
a(n) = 5*a(n-1) - 6*a(n-2).
Sum_{j=0..n-1} a(j) = (1/2)*(3^n - 1) + (2^n - 1). [Jolley] - Gary W. Adamson, Dec 20 2006
Equals double binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
If p[i] = Fibonacci(2i-5) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n) = 2*a(n-1) + 3^(n-1), with a(0)=2. - Vincenzo Librandi, Nov 18 2010
a(n) = A001550(n) - 1 = A000079(n) + A000244(n). - Reinhard Zumkeller, Mar 01 2012

Additional comments from Michael Somos, Jun 10 2000

A082103 Numbers n such that 3^n + 2^(n-1) is prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 10, 15, 21, 24, 36, 49, 51, 86, 116, 134, 176, 284, 345, 498, 544, 649, 844, 1051, 1171, 1384, 1497, 1514, 1638, 1856, 2860, 2890, 3235, 3584, 4047, 5990, 7729, 8935, 9907, 15241, 15864, 17629, 19264, 28239, 43730, 46879, 65379, 89468, 128787, 139976

Offset: 1

Labos Elemer, Apr 14 2003

nonn

a(51) > 2*10^5. - Robert Price, May 06 2016

			n=4: 89 = 2*2*2 + 3*3*3*3 is prime.
n=15: 3^15 + 2^14 = 14365291 is prime.

Cf. A082101, A082102, A095906, A093717, A093793, A093765, A096185, A093794, A093795, A096186.

[n: n in [0..1000] | IsPrime(3^n + 2^(n-1))]; // Vincenzo Librandi, Mar 20 2015

Do[ If[ PrimeQ[3^n + 2^(n - 1)], Print[n]], {n, 7650}] (* Robert G. Wilson v, May 24 2004 *)
Select[Range[6000], PrimeQ[3^# + 2^(#-1)] &] (* Vincenzo Librandi, Mar 20 2015 *)

is(n)=isprime(3^n+2^(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

Edited and extended by Robert G. Wilson v, May 25 2004
Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar
a(37)-a(50) from Robert Price, May 06 2016

A094473 Smallest prime factor of 2^n+3^n.

Original entry on oeis.org

5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 97, 5, 13, 5, 3041, 5, 13, 5, 41, 5, 13, 5, 17, 5, 13, 5, 97, 5, 13, 5, 1153, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 89, 5, 13, 5, 193, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5, 41, 5, 13, 5, 769, 5, 13, 5, 97, 5, 13, 5, 17, 5, 13, 5

Offset: 1

Labos Elemer, Jun 02 2004

nonn

If n = 4*k+1 or 4*k+3 then 2^n+3^n is divisible by 5.
If n = 4*k+2 then 2^n+3^n is divisible by 13.
Case n = 4*k including especially n = 2^j cannot be discussed with elementary tools and primality of 2^n+3^n remains open.
a(n) = 17 for n == 8 (mod 16). - Bruno Berselli, Dec 23 2019

Antti Karttunen, Table of n, a(n) for n = 1..1023

Cf. A007689, A020639, A050244, A082101, A094474-A094494.

List([1..80],n->Factors(2^n+3^n)[1]); # Muniru A Asiru, Nov 01 2018

[Min(PrimeFactors(2^n+3^n)): n in[1..100]]; // Vincenzo Librandi, Dec 23 2019

[PrimeFactors(2^n+3^n)[1]: n in[1..600]]; // Bruno Berselli, Dec 23 2019

mif[x_]:=Part[Flatten[FactorInteger[x]], 1] Table[mif[2^w+3^w], {w, 1, 75}]
FactorInteger[#][[1,1]]&/@Table[2^n+3^n,{n,80}] (* Harvey P. Dale, Mar 26 2019 *)

a(n)=factor(2^n+3^n)[1,1] \\ Charles R Greathouse IV, Apr 29 2015

A094473(n) = { my(k=(2^n+3^n)); forprime(p=2,k,if(!(k%p),return(p))); }; \\ Antti Karttunen, Nov 01 2018

a(n) = A020639(A007689(n)). - Antti Karttunen, Nov 01 2018

A094475 Primes of form 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 641

Offset: 1

Labos Elemer, Jun 01 2004

2^n+p^n is prime if n=0;or n=1 and p is a smaller of twin primes; or n=2 and 4+p^2 is prime; or n=3 and 8+p^3 is prime etc. Several conditions have to be satisfied to get a modest number of terms...

n must be zero or a power of two. Checked n being powers of two through 2^22. Thus a(5) > 10^5800000. Primes of this magnitude are rare (about 1 in 13.4 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, May 02 2013

			For n=4, p=2^4+5^4=641, so p can be prime even when the exponent is not a prime.

Cf. A094473, A094474, A082101, A094476.

[ a: n in [0..2100] | IsPrime(a) where a is 5^n+2^n]; // Vincenzo Librandi, Nov 18 2010

Select[Table[2^n+5^n,{n,0,5000}],PrimeQ] (* Harvey P. Dale, May 28 2014 *)

A294132 Sorted list of prime factors of numbers of the form 3^(2^m) + 2^(2^m) with m >= 0.

Original entry on oeis.org

5, 13, 17, 97, 257, 401, 769, 1153, 3041, 14177, 65537, 286721, 1810433, 2752513, 4043777, 7340033, 13631489, 23068673, 72222721, 319291393, 348061697, 625483777, 3937533953, 54498164737, 106216554497, 121899667073, 151597350913, 342456532993

Offset: 1

Arkadiusz Wesolowski, Oct 23 2017

nonn

Primes p such that the multiplicative order of 3/2 (mod p) is a power of 2.

			The first 5 such numbers are 5, 13, 97, 6817, 43112257, 1853024483819137. Their prime factorizations are (5), (13), (97), (17) (401), (14177) (3041), (1153) (1607133116929). - _N. J. A. Sloane_, Oct 29 2017

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..48
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.

Cf. A050244, A094499, A082101, A294133, A294134, A294135, A294136.

print1(5, ", "); forprime(p=13, 342456532993, z=znorder(Mod(3/2, p)); if(2^ispower(z)==z, print1(p, ", ")));

A050244 Numbers k such that 2^k + 3^k is a semiprime.

Original entry on oeis.org

3, 6, 7, 8, 10, 11, 12, 14, 16, 22, 32, 34, 38, 82, 83, 106, 128, 149, 218, 223, 334, 412, 436, 599, 647, 916, 1373, 4414, 7246, 8423, 10118, 10942, 15898, 42422, 65986

Offset: 1

Hugo Pfoertner, May 08 2003

Empirically, the smaller factor of 2^a(n) + 3^a(n) is a term of A294132 for all known terms. a(34) > 22000. - Hugo Pfoertner, Jul 29 2019
Terms for n >= 34 are probable semiprimes. - Tyler Busby, Feb 18 2023
Empirically, this sequence is a subsequence of A093641. No more terms of A093641 less than 10^5 are in this sequence. - Tyler Busby, Feb 20 2023

			a(1)=3 because 2^3 + 3^3 = 5 * 7.
a(2)=6 because 2^6 + 3^6 = 13 * 61.
a(3)=7 because 2^7 + 3^7 = 5 * 463.
a(4)=8 because 2^8 + 3^8 = 17 * 401.
a(5)=10 because 2^10 + 3^10 = 13 * 4621.
a(6)=11 because 2^11 + 3^11 = 5 * 35839.
a(7)=12 because 2^12 + 3^12 = 97 * 5521.
a(8)=14 because 2^14 + 3^14 = 13 * 369181.
a(9)=16 because 2^16 + 3^16 = 3041 * 14177.
a(10)=22 because 2^22 + 3^22 = 13 * 2414250301.
a(11)=32 because 2^32 + 3^32 = 1153 * 1607133116929.
a(12)=34 because 2^34 + 3^34 = 13 * 1282861452271981.
a(13)=38 because 2^38 + 3^38 = 13 * 103911691734684541.
a(14)=82 because 2^82 + 3^82 = 13 * 102329189594547549657540565413396038701.
a(15)=83 because 2^83 + 3^83 = 5 * 798167678837469920188160718521149336927.
a(16)=106 because 2^106 + 3^106 = 13 * 28900785585664327723593061693364968422740414514061.
a(17)=128 because 2^128 + 3^128 = 257 * 45876204582640401445607833244277975113391731388650867226881.
a(18)=149 because 2^149 + 3^149 = 5 * 24665899002341798194980052306171212216360861465143461865961807325057879.

factordb, Status of (3^42422+2^42422)/13.

Cf. A001358, A007689, A082101, A294132.

Corrected and extended by Hugo Pfoertner, May 12 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 30 2007
a(28)-a(32) from Sean A. Irvine, Nov 05 2009
a(33) from Hugo Pfoertner, Jul 29 2019
a(34)-a(35) from Tyler Busby, Jan 14 2023

A283653 Numbers k such that 3^k + (-2)^k is prime.

Original entry on oeis.org

0, 2, 3, 4, 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

Juri-Stepan Gerasimov, Mar 12 2017

Numbers j such that both 3^j + (-2)^j and 3^j + (-4)^j are primes: 0, 3, 4, 17, 59, ...
See Michael Somos comment in A082101.
Probably this is just A057468 with 0,2,4 added, because we already know that if another even number belong to this sequence it must be greater than log_3(10^16000000) = about 3.3*10^7. This is because 3^n+2^n can be a prime with n>0 only if n is a power of 2. - Giovanni Resta, Mar 12 2017

			4 is in this sequence because 3^4 + (-2)^4 = 97 is prime.

Cf. A174326. Subsequence of A087451. Supersequence of A057468.

Cf. A082101.

[n: n in [0..1000] | IsPrime(3^n+(-2)^n)];

Select[Range[0, 10000], PrimeQ[3^# + (-2)^#] &] (* G. C. Greubel, Jul 29 2018 *)

is(n)=isprime(3^n+(-2)^n) \\ Charles R Greathouse IV, Mar 16 2017

A094476 Primes of form 2^j + 17^j.

Original entry on oeis.org

2, 19, 293, 83537

Offset: 1

Labos Elemer, Jun 01 2004

nonn

The number j must be zero or a power of 2. Checked j being powers of two through 2^20. Thus a(5) > 10^2500000. Primes of this magnitude are rare (about 1 in 5.9 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 29 2013

			j=0: p=1+1=2;j=1: p=2+17=19;j=2: p=4+289=293;j=4: p=16+83521=83537; the j exponents are powers of 2.

Cf. A082101, A094473-A094480.

Select[Table[2^n+17^n,{n,0,2000}],PrimeQ] (* Harvey P. Dale, Nov 27 2012 *)

A094488 Primes p such that 2^j+p^j are primes for j=0,1,2,8.

Original entry on oeis.org

137, 2087, 2687, 16067, 24107, 29207, 154787, 155537, 223007, 331907, 427877, 662897, 708137, 769997, 802127, 849047, 869597, 891887, 1031117, 1068497, 1261487, 1336337, 1712567, 1794677, 1807997, 1838297, 1990577, 2189987

Offset: 1

Labos Elemer, Jun 01 2004

nonn

			For j=0 1+1=2 is prime; also terms should be lesser-twin-primes
because of p^1+2^1=p+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=256+p^8.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A082101, A094473-A094487.

{ta=Table[0, {100}], u=1}; Do[s0=2;s1=Prime[j]+2;s2=4+Prime[j]^2;s8=256+Prime[j]^8; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s8], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[200000]],AllTrue[{#+2,#^2+4,#^8+256},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 03 2018 *)

A094491 Primes p such that 2^j+p^j are primes for j=0,4,8,128.

Original entry on oeis.org

223, 2104547, 2403689, 4268233, 17620457, 21848647, 23487311, 29205821, 42889591, 43458859, 47899487, 48309017, 54666847, 61227457, 73038689, 81742547, 83574457, 85031153, 87285403, 95017003, 100339517, 103136867

Offset: 1

Labos Elemer, Jun 01 2004

nonn

Primes of 2^j+p^j form are a generalization of Fermat-primes. This is strongly supported by the observation that corresponding j-exponents are apparently powers of 2 like for the 5 known Fermat primes. See A094473-A094490.

			For j=0 1+1=2 is prime; other conditions are: because of p^4+16==prime; 3rd and 4th conditions are as follows: prime=p^8+256 and prime=2^128+p^128.

Cf. A082101, A094473-A094490.

{ta=Table[0, {100}], u=1}; Do[s0=2;s4=16+Prime[j]^4;s8=256+Prime[j]^8;s128=2^128+Prime[j]^128 If[PrimeQ[s0]&&PrimeQ[s4]&&PrimeQ[s8]&&PrimeQ[s128], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]

a(5)-a(22) from Donovan Johnson, Oct 12 2008

cp's OEIS Frontend

A007689 a(n) = 2^n + 3^n.

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A082103 Numbers n such that 3^n + 2^(n-1) is prime.

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A094473 Smallest prime factor of 2^n+3^n.

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A094475 Primes of form 2^n + 5^n.

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A294132 Sorted list of prime factors of numbers of the form 3^(2^m) + 2^(2^m) with m >= 0.

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A050244 Numbers k such that 2^k + 3^k is a semiprime.

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A283653 Numbers k such that 3^k + (-2)^k is prime.

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