Scarce total number

The totient of a number is defined in number theory as , which is also called Euler's Phi function and indicates how many coprime natural numbers there are that are not greater than . A sparse totiente number (from the English sparsely totient number ) is a natural number for which all true: ${\displaystyle k}$ ${\displaystyle \varphi (k)}$ ${\displaystyle k}$ ${\displaystyle x}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle m>k}$

${\displaystyle \varphi (m)>\varphi (k)}$.

In other words, if the totient of all numbers are greater than the totient of , then the totient number is sparse. ${\displaystyle \varphi (m)}$ ${\displaystyle m>k}$ ${\displaystyle k}$ ${\displaystyle k}$

These numbers were first mentioned by David Masser and Peter Shiu in 1986.

Examples

• The Totienten , so the number of prime natural numbers , are (for ): ${\displaystyle \varphi (k)}$ ${\displaystyle k}$ ${\displaystyle x<k}$ ${\displaystyle k=1,2,3,\ldots }$

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44, ... (sequence A000010 in OEIS )

Example :

The number is in the 8th position of the list above . The number has coprime numbers that are less than , namely and . Hence actually . ${\displaystyle 4}$ ${\displaystyle k=8}$ ${\displaystyle 4}$ ${\displaystyle k=8}$ ${\displaystyle 1,3,5}$ ${\displaystyle 7}$ ${\displaystyle \varphi (8)=4}$

In the 7th position of the list above is the number . The number is a prime number and thus has relatively prime numbers that are smaller than , namely all numbers from to . So is . ${\displaystyle 6}$ ${\displaystyle k=7}$ ${\displaystyle 6}$ ${\displaystyle k=7}$ ${\displaystyle 1}$ ${\displaystyle 6}$ ${\displaystyle \varphi (7)=6}$

• The number is not a sparse totient number: ${\displaystyle k=22}$

The totient of the number is because too precisely coprime numbers exist which are smaller than (namely ). For all larger numbers should apply. But this is not the case because, for example, the number has exactly coprime numbers (namely ) , which means that the requirement for sparse totient numbers is not fulfilled. ${\displaystyle k=22}$ ${\displaystyle \varphi (22)=10}$ ${\displaystyle k=22}$ ${\displaystyle 10}$ ${\displaystyle k=22}$ ${\displaystyle x\in \{1,3,5,7,9,13,15,17,19,21\}}$ ${\displaystyle m>22}$ ${\displaystyle \varphi (m)>\varphi (22)}$ ${\displaystyle k=24}$ ${\displaystyle 8}$ ${\displaystyle x\in \{1,5,7,11,13,17,19,23\}}$ ${\displaystyle \varphi (24)=8<10=\varphi (22)}$

• The number is a sparse totient number: ${\displaystyle k=30}$

The totient of the number is because too precisely coprime numbers exist which are smaller than (namely ). There is actually no number that has a totient that is less than or equal to . This is a sparse totient number. ${\displaystyle k=30}$ ${\displaystyle \varphi (30)=8}$ ${\displaystyle k=30}$ ${\displaystyle 8}$ ${\displaystyle k=30}$ ${\displaystyle x\in \{1,7,11,13,17,19,23,29\}}$ ${\displaystyle m>30}$ ${\displaystyle \varphi (30)=8}$ ${\displaystyle k=30}$

• The following numbers are the smallest sparsely totient numbers:

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, ... (sequence A036913 in OEIS )

• A table follows, from which one can read the sparse totient numbers a little easier. In the first column are the ascending totients , in the second column are those numbers whose totient is and in the third column you can read off the highest value in the second column. If this number is greater than all previous values, it is a sparse totient number and is colored yellow ( except for odd totients , they do not exist and are therefore omitted): ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n=1}$

properties

• Any sparse totient number is an even number.

Proof:

The proof is a proof by contradiction , an assumption is made which turns out to be false in the course of the proof. The assumption must be dropped and the opposite of the assumption must be true:

Suppose there is a sparse totient number that is odd. Be their totient . According to the definition of the sparse totient numbers, there must not be a larger number that has the same totient (it should be). ${\displaystyle k}$ ${\displaystyle \varphi (k)=n}$ ${\displaystyle m>k}$ ${\displaystyle \varphi (m)=n=\varphi (k)}$ ${\displaystyle \varphi (m)>n=\varphi (k)}$

Be . Because the odd number to be prime, and due to the calculation rules of the Euler phi function is . Thus there is a larger, even number whose totient is. This means that there can not be a sparse totient number. The assumption that an odd sparse totient number is a must be dropped, so there are no odd sparse totient numbers, all sparse totient numbers have to be even. ${\displaystyle m:=2k}$ ${\displaystyle k}$ ${\displaystyle 2}$ ${\displaystyle \varphi (m)=\varphi (2k)=\varphi (2)\cdot \varphi (k)=1\cdot \varphi (k)=\varphi (k)=n}$ ${\displaystyle 2k}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle \Box }$

• Let be an odd prime number. Then: ${\displaystyle p\in \mathbb {P} }$

Any prime faculty and is a sparse totient number. ${\displaystyle p\#}$ ${\displaystyle p\cdot p\#}$

Could this set David Masser and Peter Shiu prove.

See also

• Euler's phi function
• Highly quoted number
• Highly touted number
• High composite number
• Non-cotient
• Nontotient
• Perfect total number

Web links

• David Masser , Peter Shiu : On sparsely totient numbers. Pac. J. Math. 121 , 1986, pp. 407-426 , accessed March 1, 2020 . 
• Roger C. Baker, Glyn Harman: Sparsely totient numbers. Ann. Fac. Sci. Toulouse , VI. Sér., Math. 5 (2), 1996, pp. 183-190 , accessed March 1, 2020 . 

Individual evidence

1. ^ ^{A b} David Masser , Peter Shiu : On sparsely totient numbers. Theorem 1. Pac. J. Math. 121 , 1986, pp. 407-426 , accessed March 1, 2020 . 
2. ^ Roger C. Baker, Glyn Harman: Sparsely totient numbers. Theorem. Ann. Fac. Sci. Toulouse , VI. Sér., Math. 5 (2), 1996, pp. 183-190 , accessed March 1, 2020 . 
3. ↑ Comments on OEIS A006511
4. ↑ Comments on OEIS A036913