A247601 Least positive integer m with pi(m*n) = phi(m), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

2, 1, 13, 31, 73, 181, 443, 2249, 238839, 6473, 30001, 40123, 108539, 251707, 637321, 7554079, 4124437, 241895689, 27067097, 69709723, 179992919, 1019958623, 1208198863, 3140421743, 8179002173

Offset: 1

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

• nonn
• more

Comments

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

This is motivated by Golomb's result that for any n > 1 there is a positive integer m with mn/pi(mn) = n (i.e., pi(mn) = m).

Examples

			a(3) = 13 since pi(3*13) = 12 = phi(13).
		

Links

• S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Crossrefs

Cf. A000010, A000720, A247600, A247602, A247603, A247604, A247672, A247673.

Programs

Mathematica

Do[m=1;Label[aa];If[PrimePi[n*m]==EulerPhi[m],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,1,18}]
Table[m = 1;
While[PrimePi[n*m] != EulerPhi[m], m++]; m, {n,1,12}] (* Robert Price, Sep 08 2019 *)

Extensions

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