A020488 Numbers n such that tau(n) (or sigma_0(n)) = phi(n).

1, 3, 8, 10, 18, 24, 30

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• fini
• full

Comments

Numbers satisfying A000005(n) = A000010(n).

This sequence is complete because tau(n) < n^(2/3) for all n except a few small numbers, whereas phi(n) > n/(exp(gamma) * log(log(n)) + 3/(log(log(n))) for n > 2. log(log(n)) grows slowly, so phi(n) > tau(n) for all n greater than some relatively small constant. - Jud McCranie, Jun 17 2005

Subset of A112587. - Reinhard Zumkeller, Sep 14 2005

A. P. Minin proved in 1894 that these are the only terms. - Amiram Eldar, May 14 2017

Examples

			10 has four divisors: 1, 2, 5, 10, so tau(10) = 4. And four numbers less than 10 are coprime to 10: 1, 3, 7, 9, so phi(10) = 4. Since tau(10) = phi(10), 10 is in the sequence.
phi(12) = 4 also, but 12 has more than four divisors: 1, 2, 3, 4, 6, 12. So 12 is not in the sequence.
		

References

• L. E. Dickson, History of the Theory of Numbers, Vol. 1, (1919), Chapter X, p. 313.
• Jean-Marie De Koninck, Those Fascinating Numbers, translated by the author. Providence, Rhode Island (2009) American Mathematical Society, p. 3.
• G. Pólya and G. Szegő, Problems and Theorems in Analysis II, Springer, 1976, Part VIII, Problem 45.

Links

• A. P. Minin, On integers N such that the number of divisors of N equals the number of integers less than N and prime to it, Math. Soc. Moscow, Vol. 17, (1894), pp. 537-544 (some front matter is in English and German, article is in Russian)

Crossrefs

Cf. A064374, A064375, A064376, A064377, A000005, A000010.

Cf. A112954, A062516, A063469, A063470.

Programs

GAP

Filtered([1..1000],n->Tau(n)=Phi(n)); # Muniru A Asiru, Dec 20 2018

Magma

[n: n in [1..1000] | EulerPhi(n) eq NumberOfDivisors(n)]; // Marius A. Burtea, Dec 20 2018

Maple

select(k->tau(k)=phi(k),[$1..1000]); # Peter Luschny, Aug 26 2011

Mathematica

k = 1; s = Select[Range[100000], Equal[Sign[DivisorSigma[k - 1, #] - EulerPhi[#]^k ], 0 ] &]
Select[Range[1000], DivisorSigma[0, #] == EulerPhi[#] &] (* Alonso del Arte, Jan 15 2019 *)

PARI

isok(n) = numdiv(n) == eulerphi(n); \\ Michel Marcus, May 14 2017