A248052 Least positive integer m such that m + n divides pi(m^2) + pi(n^2), where pi(x) denotes the number of primes not exceeding x.

1, 2, 1, 8, 7, 7, 5, 1, 25, 35, 15, 14, 13, 12, 1, 4, 23, 532, 22, 385, 113, 1, 17, 138, 8, 92, 80, 44, 116, 128, 586, 165, 5, 464, 10, 39, 80, 38, 1, 52, 33, 118, 6, 28, 11, 1239, 47, 92, 517, 3, 145, 40, 8, 184, 104, 104, 16, 73, 53, 52, 5, 145, 172, 68, 11

Offset

1,2

Comments

Conjecture: (i) a(n) exists for any n > 0.

(ii) For each integer m > 0, there is a positive integer n such that m + n divides prime(m^2) + prime(n^2).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Example

a(4) = 8 since 8 + 4 = 12 divides pi(8^2) + pi(4^2) = 18 + 6 = 24.

Mathematica

Do[m=1; Label[aa]; If[Mod[PrimePi[m^2]+PrimePi[n^2], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 70}]
a[ n_] := If[ n<1, 0, Module[ {m=1}, While[ Not @ Divisible[ PrimePi[m^2] + PrimePi[n^2], m + n], m++]; m]]; (* Michael Somos, Sep 30 2014 *)

Crossrefs

Cf. A000040, A000720, A247824, A247975, A248044, A248054.

Sequence in context: A168142 A081800 A254180 * A340629 A105672 A338249

Adjacent sequences: A248049 A248050 A248051 * A248053 A248054 A248055

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2014

Status

approved