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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.
3
11, 13, 17, 19, 41, 43, 101, 103, 223, 293, 313, 331, 359, 401, 409, 439, 491, 521, 523, 571, 613, 677, 709, 821, 883, 947, 1009, 1039, 1061, 1193, 1283, 1291, 1303, 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
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}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 29 2015
STATUS
approved