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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

%I #27 Mar 08 2023 05:27:53

%S 6,2,2,23,3,1,3,1033,2,6449,15887,1,100169,268393,636917,2113589,

%T 70324093,1,27852457,78848479,2,468329417,4,1,10220118551

%N 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.

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

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685 [math.NT], 2014-2017.

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

%t Table[m = 1; While[PrimePi[n*m] != EulerPhi[m] + EulerPhi[n], m++]; m, {n, 1, 12}] (* _Robert Price_, Sep 08 2019 *)

%o (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

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

%K nonn,more

%O 1,1

%A _Zhi-Wei Sun_, Sep 22 2014

%E a(19)-a(25) from _Hiroaki Yamanouchi_, Oct 04 2014

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)