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A068145 Primes of the form a^a + b^b where a and b are positive integers. 12

%I

%S 2,5,31,257,283,823547,823799,10000823543,11112006825558043,

%T 437893890380859631,39346408075296538398967,

%U 20880467999847912043271133358823,88817841970012523233890533447265881

%N Primes of the form a^a + b^b where a and b are positive integers.

%C The sum of the reciprocals of this sequence converges to 0.73968511225249255023367393935203659031815678811682494308673702866... The PARI program below for powerpp(60) and powerpp(70) give this result for 100 digits. Is this number irrational? Transcendental? - _Cino Hilliard_, Dec 14 2002

%C Note that 3 is also a prime of the form a^a + b^b where a = 2 and b = -1. But this sequence focuses on the positive values of a and b. - _Altug Alkan_, Jan 08 2016

%H Charles R Greathouse IV, <a href="/A068145/b068145.txt">Table of n, a(n) for n = 1..94</a>

%e 257 = 4^4 + 1^1 is a prime. 823799 = 4^4 + 7^7 is a prime.

%p k := 1; for i from 2 to 100 do for j from 1 to i-1 do a := i^i+j^j; if(isprime(a)=true) then feld[k] := a; k := k+1; end if; end do; end do; sort([seq(feld[p],p=1..k-1)]);

%t nn=100; Select[ Union[ Flatten[ Table[a^a + b^b, {a, nn}, {b, a} ]]], #<nn^nn && PrimeQ[#]& ]

%t With[{nn=30},Select[Union[Total/@Tuples[Range[nn]^Range[nn],2]],PrimeQ]] (* _Harvey P. Dale_, Apr 09 2015 *)

%o (PARI) powerpp(n) = { ct=0; sr=0; a=vector(n*n*n); for(x=1,n, for(y=x,n, v = x^x+y^y; if(isprime(v),ct+=1; a[ct] = v; \ print(x" "y" "z" "v" "ct); ); ); ); for(j=1,ct, for(k=j+1,ct, if(a[j] > a[k],tmp=a[k]; a[k]=a[j]; a[j]=tmp); ); ); for(j=1,ct, if(a[j]<>a[j+1],sr+=1.0/a[j]; print1(a[j]" ")); ); print(); print(sr); }

%o (PARI) v=[2];for(a=2,380,forstep(b=a%2+1,a-1,2,if(ispseudoprime(t=a^a+b^b),v=concat(v,t);print(a"^"a" + "b"^"b))));v \\ _Charles R Greathouse IV_, Feb 14 2011

%K nonn,nice

%O 1,1

%A _Amarnath Murthy_, Feb 23 2002

%E Edited and extended by _Robert G. Wilson v_ and _Sascha Kurz_, Mar 01 2002

%E Name clarified by _Altug Alkan_, Jan 08 2016

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Last modified July 11 01:12 EDT 2020. Contains 335600 sequences. (Running on oeis4.)