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A068145
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Primes of the form a^a + b^b where a and b are positive integers.
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12
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2, 5, 31, 257, 283, 823547, 823799, 10000823543, 11112006825558043, 437893890380859631, 39346408075296538398967, 20880467999847912043271133358823, 88817841970012523233890533447265881
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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EXAMPLE
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257 = 4^4 + 1^1 is a prime. 823799 = 4^4 + 7^7 is a prime.
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MAPLE
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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)]);
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MATHEMATICA
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nn=100; Select[ Union[ Flatten[ Table[a^a + b^b, {a, nn}, {b, a} ]]], #<nn^nn && PrimeQ[#]& ]
With[{nn=30}, Select[Union[Total/@Tuples[Range[nn]^Range[nn], 2]], PrimeQ]] (* Harvey P. Dale, Apr 09 2015 *)
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PROG
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(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); }
(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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