A037171 Numbers n such that phi(n) = pi(n), i.e., A000010(n) = A000720(n).

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

Offset: 1

Author

Naohiro Nomoto

Keywords

• easy
• nonn
• fini
• full

Comments

David W. Wilson and Jeffrey Shallit showed that 90 is the last term.

Leo Moser proved in 1951 that these are the only terms, but he missed the term 10. - Amiram Eldar, May 15 2017

phi(n) >= pi(n) for n >= 61, and phi(n) > pi(n) for n > 90. - Jonathan Sondow, Dec 02 2017

Examples

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

References

• P. Birch and D. Singmaster, An elementary number theory result, Math. Soc. Newsl., 12 (1984), 10-13.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

Links

• Leo Moser, On the equation ϕ(n) = π(n), Pi Mu Epsilon Journal. Vol. 1, No. 5 (1951), pp. 177-180.

Crossrefs

Cf. A000010, A000720, A037228, A037230.

Programs

Maple

select(x->numtheory[phi](x)=numtheory[pi](x),[$1..999]); # M. F. Hasler, Mar 30 2007

PARI

for(n=1,1e5,if(primepi(n)==eulerphi(n),print(n))) /* M. F. Hasler, Mar 30 2007 */

Formula

A037228(a(n)) = 0. - Jonathan Sondow, Dec 02 2017