OFFSET
1,1
COMMENTS
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.
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).
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..3000 from Zhi-Wei Sun)
EXAMPLE
a(1) = 11 since 11 = 2 + 9 = pi(2^2) + pi(5^2) with 11, 2 and 5 all prime.
a(60) = 2381 since 2381 = 1000 + 1381 = pi(89^2) + pi(107^2) with 2381, 89 and 107 all prime.
MATHEMATICA
f[n_]:=PrimePi[Prime[n]^2]
T[1]:={f[1]}
T[n_]:=Union[T[n-1], {f[n]}]
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}];
Label[aa]; Continue, {y, 1, 353}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 29 2015
STATUS
approved