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A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way.

%I A282533 #37 Feb 25 2019 10:27:22
%S A282533 41,89,113,137,593,857,2213
%N A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way.
%C A282533 Primes of the form 2^k + p^e in more than one way where p is an odd prime (e > 1, k > 1).
%C A282533 Prime terms in A225103.
%C A282533 29 = 2^4 + 5^2 = 2 + 3^3 is a border case not included in this sequence - _Olivier Gérard_, Feb 25 2019
%C A282533 a(8) > 10^8 if it exists. - _Robert Israel_, Feb 17 2017
%C A282533 a(8) > 10^18 if it exists. - _Charles R Greathouse IV_, Feb 19 2017
%e A282533 41 = 2^4 + 5^2 = 2^5 + 3^2.
%e A282533 89 = 2^3 + 3^4 = 2^6 + 5^2.
%e A282533 113 = 2^5 + 3^4 = 2^6 + 7^2.
%e A282533 137 = 2^7 + 3^2 = 2^4 + 11^2.
%e A282533 593 = 2^9 + 3^4 = 2^6 + 23^2.
%e A282533 857 = 2^7 + 3^6 = 2^4 + 29^2.
%e A282533 2213 = 2^4 + 13^3 = 2^2 + 47^2.
%p A282533 N:= 10^6: # to get all terms <= N
%p A282533 B:= Vector(N):
%p A282533 C:= Vector(N):
%p A282533 for k from 2 to ilog2(N) do B[2^k]:= 1 od:
%p A282533 p:= 2:
%p A282533 do
%p A282533   p:= nextprime(p);
%p A282533   if p^2 > N then break fi;
%p A282533   for k from 2 to floor(log[p](N)) do C[p^k]:= 1 od:
%p A282533 od:
%p A282533 R:= SignalProcessing:-Convolution(B,C):
%p A282533 select(t -> isprime(t) and R[t-1] > 1.5, [seq(i,i=3..N,2)]); # _Robert Israel_, Feb 17 2017
%t A282533 Select[Prime@ Range[10^3], Function[n, Count[Transpose@{n - #, #}, w_ /; Times @@ Boole@ Map[And[PrimePowerQ@ #, ! PrimeQ@ #] &, w] > 0] >= 2 &@ Range[4, Floor[n/2]]]] (* or *)
%t A282533 With[{n = 10^8}, Keys@ Select[#, Length@ # > 1 &] &@ GroupBy[#, First] &@ SortBy[Transpose@ {Map[Total, #], #}, First] &@ Select[Union@ Map[Sort, Tuples[#, 2]], PrimeQ@ Total@ # &] &@ Flatten@ Map[#^Range[2, Log[#, Prime@ n]] &, Array[Prime@ # &, Floor@ Sqrt@ n]]] (* _Michael De Vlieger_, Feb 19 2017, latter program Version 10 *)
%o A282533 (MATLAB)
%o A282533 N = 10^8; % to get all terms <= N
%o A282533 C = sparse(1,N);
%o A282533 for p = primes(sqrt(N))
%o A282533   C(p .^ [2:floor(log(N)/log(p))]) = 1;
%o A282533 end
%o A282533 R = zeros(1,N);
%o A282533 for k = 2: floor(log2(N))
%o A282533   R((2^k+1):N) = R((2^k+1):N) + C(1:(N-2^k));
%o A282533 end
%o A282533 P = primes(N);
%o A282533 P(R(P) > 1.5) % _Robert Israel_, Feb 17 2017
%o A282533 (PARI) is(n) = if(!ispseudoprime(n), return(0), my(x=n-1, y=1, i=0); while(y < x, if(isprimepower(x) > 1 && isprimepower(y) > 1, if(i==0, i++, return(1))); y++; x--)); 0 \\ _Felix Fröhlich_, Feb 18 2017
%o A282533 (PARI) has(p)=my(t,q); p>40 && sum(k=2,logint(p-9,2), t=2^k; sum(e=2,logint(p-t,3), ispower(p-t,e,&q) && isprime(q)))>1
%o A282533 list(lim)=my(v=List(),t,q); lim\=1; if(lim<9,lim=9); for(k=2,logint(lim-9,2), t=2^k; for(e=2,logint(lim-t,3), forprime(p=3,sqrtnint(lim-t,e), q=t+p^e; if(isprime(q) && has(q), listput(v,q))))); Set(v) \\ _Charles R Greathouse IV_, Feb 18 2017
%Y A282533 Cf. A225099, A225102, A225103, A246547.
%Y A282533 Cf. A115231 (prime numbers which cannot be written as 2^a + p^b, b>=0)
%K A282533 nonn,more
%O A282533 1,1
%A A282533 _Altug Alkan_, Feb 17 2017