A247672 Least integer m > 0 with pi(m*n) = phi(m) + phi(n), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

6, 2, 2, 23, 3, 1, 3, 1033, 2, 6449, 15887, 1, 100169, 268393, 636917, 2113589, 70324093, 1, 27852457, 78848479, 2, 468329417, 4, 1, 10220118551

Offset: 1

Author

Zhi-Wei Sun, Sep 22 2014

Keywords

• nonn
• more

Comments

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

Examples

			a(1) = 6 since pi(6) = 3 = phi(1) + phi(6), and pi(1*m) = phi(1) + phi(m) for no m < 6.
		

Links

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

Crossrefs

Cf. A000010, A000720, A247600, A247601, A247602, A247603, A247604, A247673.

Programs

Mathematica

Table[m = 1; While[PrimePi[n*m] != EulerPhi[m] + EulerPhi[n], m++]; m, {n, 1, 12}] (* Robert Price, Sep 08 2019 *)

Perl

use ntheory ":all"; for my $n (1..16) { my $m=1; $m++ until (prime_count($m*$n) == euler_phi($m) + euler_phi($n)); say "$n $m"; } # Dana Jacobsen, Mar 07 2023

Extensions

a(19)-a(25) from Hiroaki Yamanouchi, Oct 04 2014