In
mathematics, a sparsely totient number is a certain kind of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
. A natural number, ''n'', is sparsely totient if for all ''m'' > ''n'',
:
where
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
. The first few sparsely totient numbers are:
2,
6,
12,
18,
30,
42,
60,
66,
90,
120 120 may refer to:
*120 (number), the number
* AD 120, a year in the 2nd century AD
*120 BC, a year in the 2nd century BC
*120 film, a film format for still photography
* ''120'' (film), a 2008 film
*120 (MBTA bus)
*120 (New Jersey bus)
*120 (Kent) ...
,
126 126 may refer to:
*126 (number), a natural number
*AD 126, a year in the 2nd century AD
*126 BC, a year in the 2nd century BC
*126 film, a cartridge-based film format used in still photography
* 126 (New Jersey bus)
* 126 Artist-run Gallery
*Interst ...
,
150 150 may refer to:
*150 (number), a natural number
* AD 150, a year in the 2nd century AD
*150 BC, a year in the 2nd century BC
*150 Regiment RLC
* Combined Task Force 150
See also
* List of highways numbered 150
The following highways are number ...
,
210,
240
__NOTOC__
Year 240 ( CCXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Venustus (or, less frequently, year 993 ''Ab u ...
,
270,
330
__NOTOC__
Year 330 ( CCCXXX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gallicanus and Tullianus (or, less frequently, year 1 ...
,
420
420 may refer to:
* 420 (number)
*420 (cannabis culture), informal reference to cannabis use and celebrations on April 20
**California Senate Bill 420 or the Medical Marijuana Program Act
* AD 420, a year in the 5th century of the Julian calendar
* ...
, 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, ... .
The concept was introduced by
David Masser
David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Dioph ...
and
Peter Man-Kit Shiu in 1986. As they showed, every
primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
is sparsely totient.
Properties
* If ''P''(''n'') is the largest
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of ''n'', then
.
*
holds for an exponent
.
* It is conjectured that
.
References
*
*
{{Classes of natural numbers
Integer sequences