

A020488


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


29




OFFSET

1,2


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


REFERENCES

L. E. Dickson, History of the Theory of Numbers, Vol. 1, (1919), Chapter X, p. 313.
JeanMarie 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

Table of n, a(n) for n=1..7.
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. 537544 (some front matter is in English and German, article is in Russian)


EXAMPLE

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.


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 *)


PROG

(PARI) isok(n) = numdiv(n) == eulerphi(n); \\ Michel Marcus, May 14 2017
(MAGMA) [n: n in [1..1000]  EulerPhi(n) eq NumberOfDivisors(n)]; // Marius A. Burtea, Dec 20 2018
(GAP) Filtered([1..1000], n>Tau(n)=Phi(n)); # Muniru A Asiru, Dec 20 2018


CROSSREFS

Cf. A064374, A064375, A064376, A064377, A000005, A000010.
Cf. A112954, A062516, A063469, A063470.
Sequence in context: A244353 A143144 A261929 * A064435 A079541 A283757
Adjacent sequences: A020485 A020486 A020487 * A020489 A020490 A020491


KEYWORD

nonn,fini,full


AUTHOR

David W. Wilson


STATUS

approved



