A247600 Least positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

9, 7, 6, 998, 5, 5, 5, 5, 5, 5, 636787, 1617099, 4124188, 10553076, 5, 5, 179992154, 465769460, 1208198239, 3140421185, 5, 5, 5, 145935688930, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 5

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

• nonn
• more

Comments

The author proved that a(n) exists for every n >= 5.

a(39) = a(41) = 5. - Chai Wah Wu, Jun 06 2024

Examples

			a(5) = 9 since pi(5*9) = 14 = 5 + 9, and pi(5*m) = 5 + m for no m < 9.
		

Links

• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685 [math.NT], 2014-2017.

Crossrefs

Cf. A247601, A247602, A247603, A247604, A000720, A281197.

Programs

Mathematica

Do[m=1;Label[aa];If[PrimePi[n*m]==m+n,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,5,21}]
Table[m = 1; While[PrimePi[m*n] != m + n, m++]; m, {n, 5, 14}] (* Robert Price, Mar 20 2019 *)

Extensions

a(22)-a(37) from Chai Wah Wu, May 03 2018