A094499 -id:A094499 - 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

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, ", ")));

A094316 Primes p for which 2^j+p^j is also prime for j in {0,2,8,512}.

Original entry on oeis.org

13, 4133, 1831343, 2320583, 3828673, 9173893, 23658377, 24037537, 42489677, 56253203, 78222863, 96325093, 99846337, 110453773, 110468653, 117748427, 122173187, 130937467, 138072163, 146981537, 174978913, 184050553, 186927817

Offset: 1

Labos Elemer, Jun 02 2004

nonn

			Smallest such prime is 13 and the relevant four primes are
2, 173, 815730977 and a 571-digit prime.

Cf. A082101, A094473-A094499.

{ta=Table[0, {100}], u=1}; {exponents, {a, b, c, d}={0, 2, 8, 512}} Do[s0=Prime[j]^a+2^a;s1=Prime[j]^b+2^b;s2=Prime[j]^c+2^c;s3=Prime[j]^d+2^d; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}] ta

a(6)-a(23) from Donovan Johnson, Oct 12 2008

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A094316 Primes p for which 2^j+p^j is also prime for j in {0,2,8,512}.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions