

A123206


Primes of the form x^y  y^x, for x,y > 1.


6



7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249
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OFFSET

1,1


COMMENTS

These are the primes in A045575, numbers of the form x^y  y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k  k^n) for k>1.
If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)y^x.  M. F. Hasler, Aug 19 2014


LINKS

T. D. Noe, Table of n, a(n) for n=1..101 (terms < 10^400)
H. Lifchitz & R. Lifchitz, PRP of the form x^yy^x on primenumbers.net.


EXAMPLE

The primes 6102977801 and 1490116119372884249 are of the form 5^yy^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


MAPLE

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


MATHEMATICA

Take[Select[Intersection[Flatten[Table[Abs[x^yy^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^kk^n]; num<nn, Sow[num]; While[k++; num=n^kk^n; num<nn, Sow[num]]]][[2, 1]]]; Select[t, PrimeQ]
With[{nn=30}, Take[Sort[Select[#[[1]]^#[[2]]#[[2]]^#[[1]]&/@Subsets[ Range[ 2nn], {2}], #>0&&PrimeQ[#]&]], nn]] (* Harvey P. Dale, Nov 23 2013 *)


PROG

(PARI) a=[]; for(S=1, 199, for(x=2, S2, ispseudoprime(p=x^(y=Sx)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


CROSSREFS

Cf. A045575, A122735, A078202, A082754, A055651, A094133.
A163319 is the subsequences for fixed x=3, A243114 for x=6.
Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.
Sequence in context: A107693 A217717 A122528 * A035078 A177123 A124165
Adjacent sequences: A123203 A123204 A123205 * A123207 A123208 A123209


KEYWORD

nonn


AUTHOR

Alexander Adamchuk, Oct 04 2006


STATUS

approved



