A037171 - OEIS

%I #31 Nov 20 2023 15:28:17

%S 2,3,4,8,10,14,20,90

%N Numbers n such that phi(n) = pi(n), i.e., A000010(n) = A000720(n).

%C _David W. Wilson_ and _Jeffrey Shallit_ showed that 90 is the last term.

%C Leo Moser proved in 1951 that these are the only terms, but he missed the term 10. - _Amiram Eldar_, May 15 2017

%C phi(n) >= pi(n) for n >= 61, and phi(n) > pi(n) for n > 90. - _Jonathan Sondow_, Dec 02 2017

%D P. Birch and D. Singmaster, An elementary number theory result, Math. Soc. Newsl., 12 (1984), 10-13.

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

%H Leo Moser, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.1.No.5.pdf">On the equation ϕ(n) = π(n)</a>, Pi Mu Epsilon Journal. Vol. 1, No. 5 (1951), pp. 177-180.

%F A037228(a(n)) = 0. - _Jonathan Sondow_, Dec 02 2017

%e phi(10)=4, pi(10)=4.

%e a(1)=2 since k=2 is the lowest index for which A000720(n) = A000010(n), i.e., EulerPhi(n) = PrimePi(n). - _M. F. Hasler_, Mar 30 2007

%p select(x->numtheory[phi](x)=numtheory[pi](x),[$1..999]); # _M. F. Hasler_, Mar 30 2007

%o (PARI) for(n=1,1e5,if(primepi(n)==eulerphi(n),print(n))) /* _M. F. Hasler_, Mar 30 2007 */

%Y Cf. A000010, A000720, A037228, A037230.

%K easy,nonn,fini,full

%O 1,1

%A _Naohiro Nomoto_