A248044 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, 3, 1, 4, 12, 11, 1, 8, 7, 16, 2, 5, 26, 25, 24, 4, 228, 227, 46, 45, 44, 43, 16, 6, 5, 1, 27, 26, 45, 44, 12526, 12525, 12524, 12523, 2970, 502, 351, 350, 46, 45, 236, 235, 10, 9, 8, 4, 1078, 1077, 576, 575, 574, 198, 63, 62, 61, 176, 16, 10, 362, 70

Offset: 1

Author

Zhi-Wei Sun, Sep 30 2014

Keywords

• nonn
• look

Comments

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

Examples

			a(5) = 12 since 12 + 5 = 17 divides pi(12)^2 + pi(5)^2 = 5^2 + 3^2 = 34.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1387 from Zhi-Wei Sun)
• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Crossrefs

Cf. A000720, A247824, A247975, A248035, A248036.

Programs

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,1,60}]