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

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

Author

Labos Elemer, Jun 02 2004

Keywords

• nonn

Comments

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

Links

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

Crossrefs

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

Programs

GAP

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

Magma

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

Magma

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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