7500 (number)

7500
7500
	seven thousand five hundred
← 7498 7499 7500 7501 7502 →
Factorization	2 2 · 3 · 5 4
Roman notation	V MMD
Binary	1110101001100
Octal	16514
Hexadecimal	1D4C

7500 ( seven thousand five hundred ) is a natural number located between the numbers 7499 and 7501.

Content

In other languages

- Dunton Ngagya (Tibetan).

- tiedphan haroy (Thai).

Math

7500 is an even composite excess number, tau number ^[1] , number of harshad ^[1] , Ulam number ^[1] ^[2] , practical number ^[1] ^[2] , Zumkeller number ^[1] .

The number 7500 is a in the number system with base 7 ^[1] :

7500=30603_{7}.

{\ displaystyle 7500 = 30603_ {7}.}

7500 is the Saint Exupery number, or the product of the elements of the Pythagorean triple ^[1] ^[3] ^[4] :

7500=15\cdot 20\cdot 25,

{\ displaystyle 7500 = 15 \ cdot 20 \ cdot 25,}

15^{2}+20^{2}=25^{2}.

{\ displaystyle 15 ^ {2} + 20 ^ {2} = 25 ^ {2}.}

It is not known whether there exist two different Pythagorean triples with one product ^[3] ^[5] .

Divisibility

For the number 7500 the equality holds ^[6]

\varphi (n)=\varphi (n-\varphi (n)),

{\ displaystyle \ varphi (n) = \ varphi (n- \ varphi (n)),}

Where $\varphi (n)$ ${\ displaystyle \ varphi (n)}$ - Euler function .

7500 divides the sum of its largest common divisors with all the smaller numbers ^[7] :

\sum _{n=1}^{7500}{(n,7500)}=105\ 000=7500\cdot 14.

{\ displaystyle \ sum _ {n = 1} ^ {7500} {(n, 7500)} = 105 \ 000 = 7500 \ cdot 14.}

The product of prime divisors of the number 7500 is equal to the number of divisors of 7500 ^[8] ^[9] :

\operatorname {rad} (7500)=\sigma _{0}(7500)=30.

{\ displaystyle \ operatorname {rad} (7500) = \ sigma _ {0} (7500) = 30.}

Enumeration

There are 7500 3 × 3 matrices with coefficients from {0, 1, 2 } in which the sum of the elements of the side diagonal does not exceed the sum of the elements of the main diagonal and the sum of the elements of the second column does not exceed the sum of the elements of the second row ^[10] ^[11] ^[12] . The number of 4 × 4 matrices for each of the 3 × 3 submatrices of which the indicated conditions are satisfied is 567 009 ^[10] ^[12] .

Group Theory

7500 is the order of some finite ^[13] .

Collatz Hypothesis

The Syracuse sequence , starting with the number 7500, in 88 steps reaches a maximum ^[14] , equal to 250 504 ^[15] . After reaching the maximum, 88 more steps are required to reach the unit ^[16] . The number 7500 is the twentieth natural number for which the number of steps to the maximum is equal to the number of steps from the maximum to one ^[17] .

Timeline

7500 year. The Byzantine era of the year 7500 was celebrated on September 14, 1991 .

Isopssephia

(hunk) - incense. See Armenian isopssephia .

Astronomy

NGC 7500

Notes

↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Giovanni Resta. 7500: facts & properties (unopened) . Numbers Aplenty .
↑ ¹ ² Tanya Khovanova. 7500 (neopr.) . Number Gossip .
↑ ¹ ² Giovanni Resta. Saint-Exupery numbers (neopr.) . Numbers Aplenty .
↑ Sequence A057096 in OEIS = Saint-Exupery numbers: ordered products of the three sides of Pythagorean triangles
↑ Weisstein, Eric W. Pythagorean Triple on the Wolfram MathWorld website.
↑ Sequence A051487 in OEIS // Fragment: 6144, 6348, 6930, 7500 , 7986, 9240, 9600
↑ Sequence A066862 in OEIS // Fragment: 6400, 6912, 7056, 7500 , 8775, 9216, 11 520
↑ Sequence A070226 in OEIS // Fragment: 4032, 4050, 5625, 7500 , 9408, 11 250, 11 264
↑ Sequence A120737 in OEIS // Fragment: 6144, 6561, 6912, 7500 , 7776, 8192, 8748
↑ ¹ ² A258604 sequence in OEIS
↑ A258605 sequence in OEIS
↑ ¹ ² A258612 sequence in OEIS
↑ Sequence A060793 in OEIS = Orders of finite perfect groups (groups such that G = G 'where G' is the commutator subgroup of G) // Fragment: 6072, 6840, 7200, 7500 , 7560, 7680, 7800
↑ A220421 sequence in OEIS = Number of halving and tripling steps to reach the largest value in the Collatz (3x + 1) trajectory of n
↑ A025586 sequence in OEIS = Largest value in `3x + 1 'trajectory of n
↑ A222641 sequence in OEIS = Number of iterations in Collatz (3x + 1) trajectory of n to reach 1 from the highest term
↑ Sequence A224303 in OEIS = Numbers n for which number of iterations to reach the largest equals number of iterations to reach 1 from the largest in Collatz (3x + 1) trajectory of n

Links

The number 7500 in OEIS

[nap-1] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Giovanni Resta. 7500: facts & properties (unopened) . Numbers Aplenty .

[gossip-2] ¹ ² Tanya Khovanova. 7500 (neopr.) . Number Gossip .

[nap-se-3] ¹ ² Giovanni Resta. Saint-Exupery numbers (neopr.) . Numbers Aplenty .

[oeis-a057096-4] Sequence A057096 in OEIS = Saint-Exupery numbers: ordered products of the three sides of Pythagorean triangles

[5] Weisstein, Eric W. Pythagorean Triple on the Wolfram MathWorld website.

[6] Sequence A051487 in OEIS // Fragment: 6144, 6348, 6930, 7500 , 7986, 9240, 9600

[7] Sequence A066862 in OEIS // Fragment: 6400, 6912, 7056, 7500 , 8775, 9216, 11 520

[8] Sequence A070226 in OEIS // Fragment: 4032, 4050, 5625, 7500 , 9408, 11 250, 11 264

[9] Sequence A120737 in OEIS // Fragment: 6144, 6561, 6912, 7500 , 7776, 8192, 8748

[oeis-a258604-10] ¹ ² A258604 sequence in OEIS

[11] A258605 sequence in OEIS

[oeis-a258612-12] ¹ ² A258612 sequence in OEIS

[13] Sequence A060793 in OEIS = Orders of finite perfect groups (groups such that G = G 'where G' is the commutator subgroup of G) // Fragment: 6072, 6840, 7200, 7500 , 7560, 7680, 7800

[14] A220421 sequence in OEIS = Number of halving and tripling steps to reach the largest value in the Collatz (3x + 1) trajectory of n

[15] A025586 sequence in OEIS = Largest value in `3x + 1 'trajectory of n

[16] A222641 sequence in OEIS = Number of iterations in Collatz (3x + 1) trajectory of n to reach 1 from the highest term

[17] Sequence A224303 in OEIS = Numbers n for which number of iterations to reach the largest equals number of iterations to reach 1 from the largest in Collatz (3x + 1) trajectory of n