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

1,1

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).

Links

Table of n, a(n) for n=1..25.

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.

Example

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

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 *)

Crossrefs

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

Sequence in context: A264373 A217490 A317384 * A013020 A012906 A013017

Adjacent sequences: A247598 A247599 A247600 * A247602 A247603 A247604

Keyword

nonn,more

Author

Zhi-Wei Sun, Sep 21 2014

Extensions

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

Status

approved