A094474 -id:A094474 - cp's OEIS frontend

A082101 Primes of form 2^k + 3^k.

2, 5, 13, 97

Offset: 1

Labos Elemer, Apr 14 2003

nonn

Next term, if it exists, is > 10^125074. - David Wasserman, Aug 13 2004
Since x+y is a factor of x^m+y^m if m is odd, 2^m+3^m is divisible by 2+3=5 unless m is zero or a power of 2. This is similar to Fermat numbers 1+2^m. - Michael Somos, Aug 27 2004
Checked k being powers of two through 2^21. Thus a(5) > 10^2000000. Primes of this magnitude are rare (about 1 in 4.6 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 25 2013
If a(5) exists it is greater than 10^16000000. Probably complete. - Charles R Greathouse IV, Apr 29 2013

			m=0: 1+1, m=1: 2+3, m=2: 4+9, m=4: 16+81.

Cf. A094474-A094499.

a={}; Do[If[PrimeQ[p=2^n+3^n], AppendTo[a, p]], {n, 0, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Table[2^k+3^k,{k,0,100}],PrimeQ] (* Harvey P. Dale, May 14 2014 *)

print1(2);for(n=0,99,if(ispseudoprime(t=2^(2^n)+3^(2^n)),print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

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 *)

A337429 a(n) is the largest prime factor of 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 19, 641, 41, 541, 1597, 22993, 397, 5521, 303293, 380881, 25117, 210466621, 508771, 1573071713, 108991369171, 1343341, 2724783836059, 39558401, 2525293, 4807441, 215038823, 1173553, 61001, 16463734208221, 3813697527769, 58116853330557841, 327866809, 99901

Offset: 0

Hugo Pfoertner, Sep 23 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 0..387 (terms 0..200 from Hugo Pfoertner)

Cf. A006530, A074600, A094474, A094475, A122118.

a[n_] := FactorInteger[2^n + 5^n][[-1, 1]]; Array[a, 31, 0] (* Amiram Eldar, Mar 30 2023 *)

for(n=0,30,my(p=2^n+5^n);print1(vecmax(factor(p)[,1]),", "))

a(n) = A006530(A074600(n)).

A094477 Primes of form 2^n + 37^n.

Original entry on oeis.org

2, 1373, 1874177, 23169162752708970943114627382699355445603465075569066753527132965271355336698663708393617779709970177

Offset: 1

Labos Elemer, Jun 01 2004

nonn

The number n must be zero or a power of 2. Checked n being powers of two through 2^20. Thus a(5) > 10^3200000. Primes of this magnitude are rare (about 1 in 7.5 million), so chance of finding one is remote with today's computer algorithms and speeds. - Robert Price, Apr 29 2013

Cf. A094473, A094474, A082101, A094476.

No more terms for n < 1000, so the next term will be too large to include. - Hugo Pfoertner, Aug 17 2004

A094498 Least prime factor of 2^(4n) + 3^(4n) = 16^n + 81^n.

Original entry on oeis.org

97, 17, 97, 3041, 41, 17, 97, 1153, 97, 17, 89, 193, 97, 17, 41, 769, 97, 17, 97, 3041, 97, 17, 97, 1153, 41, 17, 97, 3041, 97, 17, 97, 257, 89, 17, 41, 193, 97, 17, 97, 1153, 97, 17, 97, 353, 41, 17, 97, 769, 97, 17

Offset: 1

Labos Elemer, Jun 02 2004

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..255

Cf. A082101, A094474-A094494.

mif[x_]:=Part[Flatten[FactorInteger[x]], 1] Table[mif[16^w+81^w], {w, 1, 20}]
Table[FactorInteger[16^n+81^n][[1,1]],{n,50}] (* Harvey P. Dale, Jun 02 2014 *)

a(n) = vecmin(factor(16^n + 81^n)[,1]); \\ Michel Marcus, Oct 15 2019

More terms from Harvey P. Dale, Jun 02 2014

Name corrected by Chai Wah Wu, Oct 14 2019

cp's OEIS Frontend

A082101 Primes of form 2^k + 3^k.

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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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A337429 a(n) is the largest prime factor of 2^n + 5^n.

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A094477 Primes of form 2^n + 37^n.

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A094498 Least prime factor of 2^(4*n) + 3^(4*n) = 16^n + 81^n.

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A094498 Least prime factor of 2^(4n) + 3^(4n) = 16^n + 81^n.