Number n where phi(m) is greater than phi(n) for all m greater than n
In mathematics , specifically number theory , a sparsely totient number is a natural number , n , such that for all m > n ,
φ
(
m
)
>
φ
(
n
)
{\displaystyle \varphi (m)>\varphi (n)}
where
φ
{\displaystyle \varphi }
Euler's totient function . The first few sparsely totient numbers are:
2 , 6 , 12 , 18 , 30 , 42 , 60 , 66 , 90 , 120 , 126 , 150 , 210 , 240 , 270 , 330 , 420 , 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660, 9870, ... (sequence A036913 OEIS ).
The concept was introduced by David Masser and Peter Man-Kit Shiu in 1986. As they showed, every primorial is sparsely totient.
If P (n ) is the largest prime factor of n , then
lim inf
P
(
n
)
/
log
n
=
1
{\displaystyle \liminf P(n)/\log n=1}
P
(
n
)
≪
log
δ
n
{\displaystyle P(n)\ll \log ^{\delta }n}
δ
=
37
/
20
{\displaystyle \delta =37/20}
It is conjectured that
lim sup
P
(
n
)
/
log
n
=
2
{\displaystyle \limsup P(n)/\log n=2}
They are always even because x is odd, then 2x also has the same Totient function, trivially failing the condition that all numbers more than it has more value of Totient function than it.
Possessing a specific set of other numbers
Expressible via specific sums
