|
|
A020488
|
|
Numbers n such that tau(n) (or sigma_0(n)) = phi(n).
|
|
32
|
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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
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.
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,fini,full
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|