This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(9) = 177 because we can write 177 = 2^7 + 7^2.
N:= 10^7: # to get all terms <= N
A:= {3}:
for n from 2 to floor(N^(1/2)) do
for k from 2 do
a:= n^k + k^n;
if a > N then break fi;
A:= A union {a};
od
od:
A; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 13 2015
Take[Sort[Flatten[Table[x^y + y^x, {x, 2, 100}, {y, x, 100}]]], 42] (* Alonso del Arte, Apr 05 2006 *)
nn=10^50; n=1; Union[Reap[While[n++; num=2*n^n; num
The least x such that x^y + y^x is prime for some y>1, y<x is a(1)=3, the smallest such y is a(1)=2, yielding the prime A162490(1) = 9 + 8 = 17. The least x > a(4)=21 such that x^y + y^x is prime for some y<x, y>1, is a(5)=24, yielding the prime A162490(5) for y=A162489(5)=5, while A162486(5)=33, yielding the smaller prime A094133(5)=8589935681 with y=A162487(5), comes only after a(6)=32.
lst = {}; Do[ If[ PrimeQ[x^y + y^x], AppendTo[lst, x]], {x, 3, 680}, {y, 2, x - 1}]; Union@ lst (* Robert G. Wilson v, Aug 17 2009 *)
for(i=3,999,for(j=2,i-1,is/*pseudo*/prime(i^j+j^i)|next;print1(i", ");break))
a(3) = 593 = 2^9 + 9^2. a(4) = 32993 = 2^15 + 15^2.
Select[Table[n^2+2^n,{n,1000}],PrimeQ] (* Harvey P. Dale, Jul 19 2017 *)
for(n=1,10^7, if(isprime(n^2+2^n),print(n^2+2^n)))
The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014
N:= 10^100: # to get all terms <= N
A:= NULL:
for x from 2 while x^(x+1) - (x+1)^x <= N do
for y from x+1 do
z:= x^y - y^x;
if z > N then break
elif z > 0 and isprime(z) then A:=A, z;
fi
od od:
{A}; # Robert Israel, Aug 29 2014
Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num0&&PrimeQ[#]&]],nn]] (* Harvey P. Dale, Nov 23 2013 *)
a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - M. F. Hasler, Aug 19 2014
for(n=1,5000, if(isprime(2^n+n^2),print(n)))
1^2 + 2^1 = 3 and 2^3 + 3^2 = 17 are the primes corresponding to the first two terms. The next five terms correspond to primes of 155, 870, 2487, 3678 and 9106 decimal digits.
Do[ If[ PrimeQ[n^(n + 1) + (n + 1)^n], Print[n]], {n, 1, 1650}]
for(n=1,1650, if(isprime((n^(n+1))+((n+1)^n)), print1(n,",")))
The least x such that x^y+y^x is prime for some x>y>1 is A162488(1)=3, for y=A162489(1)=2, yielding the prime a(1) = 9 + 8 = 17.
for(i=3,999,for(j=2,i-1,isprime(i^j+j^i)||next;print1(i^j+j^i", ");break))
2 is OK because 5^2 + 2^5 = 25 + 32 = 57 = 3*19 (semiprime).
IsSemiprime:=func< n|&+[k[2]: k in Factorization(n)] eq 2 >; [n: n in [1..85]|IsSemiprime(5^n+n^5)]; // Vincenzo Librandi, Dec 16 2010
Select[Range[100],PrimeOmega[5^# + #^5]==2&] (* Vincenzo Librandi, May 21 2014 *)
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