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.

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}]

Crossrefs

Cf. A000040, A000290, A000720, A237687, A262447, A262698, A262730.

Sequence in context: A275058 A157175 A132244 * A087201 A068492 A206287

Adjacent sequences: A262728 A262729 A262730 * A262732 A262733 A262734

Keyword

nonn

Author

Zhi-Wei Sun, Sep 29 2015

Status

approved