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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.
7
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
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 *)
KEYWORD
nonn,more
AUTHOR
Zhi-Wei Sun, Sep 21 2014
EXTENSIONS
a(19)-a(25) from Hiroaki Yamanouchi, Oct 04 2014
STATUS
approved