A247602 Least positive integer m with pi(m*n) = phi(m+n), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

3, 2, 1, 91, 6, 5, 1, 5, 1, 8041, 15870, 39865, 1, 251625, 637064, 1829661, 4124240, 10553093, 1, 69709253, 179992156, 465769749, 1210576800, 3140421235, 13974959892

Offset: 1

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

• nonn
• more

Comments

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

Examples

			a(1) = 3 since pi(1*3) = 2 = phi(1+3).
		

Links

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

Crossrefs

Cf. A000010, A000720, A247600, A247601, A247603, A247604, A247672, A247673.

Programs

Mathematica

Do[m=1;Label[aa];If[PrimePi[n*m]==EulerPhi[m+n],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,1,20}]
Table[m = 1;
While[PrimePi[n*m] != EulerPhi[m + n], m++]; m, {n, 1, 13}] (* Robert Price, Sep 08 2019 *)

PARI

a(n) = {my(m = 1); while (primepi(m*n) != eulerphi(m+n), m++); m;} \\ Michel Marcus, Sep 22 2014

Extensions

a(21)-a(25) from Hiroaki Yamanouchi, Oct 04 2014