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A262731 Primes p in the form pi(q^2)+pi(r^2) with q and r both prime, where pi(x) denotes the number of primes not exceeding x.

%I A262731 #14 May 31 2018 20:16:38
%S A262731 11,13,17,19,41,43,101,103,223,293,313,331,359,401,409,439,491,521,
%T A262731 523,571,613,677,709,821,883,947,1009,1039,1061,1193,1283,1291,1303,
%U A262731 1373,1409,1427,1453,1471,1487,1543,1553,1609,1669,1697,1811,1861,1879,1907,1949,1999,2039,2063,2143,2213,2239,2251,2267,2287,2309,2381
%N A262731 Primes p in the form pi(q^2)+pi(r^2) with q and r both prime, where pi(x) denotes the number of primes not exceeding x.
%C A262731 Conjecture: The sequence has infinitely many terms. In general, for each n = 2,3,4,... there are infinitely many primes p in the form pi(q^n)+pi(r^n) with q and r both prime.
%C A262731 Compare this conjecture with the well-known result that there are infinitely many primes p in the form x^2+y^2 with x and y positive integers (such a prime p is either 2 or congruent to 1 modulo 4).
%H A262731 Chai Wah Wu, <a href="/A262731/b262731.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..3000 from Zhi-Wei Sun)
%e A262731 a(1) = 11 since 11 = 2 + 9 = pi(2^2) + pi(5^2) with 11, 2 and 5 all prime.
%e A262731 a(60) = 2381 since 2381 = 1000 + 1381 = pi(89^2) + pi(107^2) with 2381, 89 and 107 all prime.
%t A262731 f[n_]:=PrimePi[Prime[n]^2]
%t A262731 T[1]:={f[1]}
%t A262731 T[n_]:=Union[T[n-1],{f[n]}]
%t A262731 n=0;Do[Do[If[f[x]>Prime[y],Goto[aa]];If[MemberQ[T[y],Prime[y]-f[x]],n=n+1;Print[n," ",Prime[y]];Goto[aa]];Continue,{x,1,y}];
%t A262731 Label[aa];Continue,{y,1,353}]
%Y A262731 Cf. A000040, A000290, A000720, A237687, A262447, A262698, A262730.
%K A262731 nonn
%O A262731 1,1
%A A262731 _Zhi-Wei Sun_, Sep 29 2015