34 (thirty-four) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
33 and preceding
35.
In mathematics
34 is the twelfth
semiprime, with four
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s including
1 and itself. Specifically, 34 is the ninth distinct
semiprime, it being the sixth of the form
. Its neighbors
33 and
35 are also distinct semiprimes with four divisors each, where 34 is the smallest number to be surrounded by numbers with the same number of divisors it has. This is the first distinct semiprime treble cluster, the next being (
85,
86,
87).
34 is the sum of the first two
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
s
6 +
28, whose difference is its
composite index (
22).
Its
reduced totient and
Euler totient values are both 16 (or 4
2 = 2
4). The sum of all its divisors aside from one equals
53, which is the sixteenth prime number.
There is no solution to the equation
φ(''x'') = 34, making 34 a
nontotient
In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotie ...
. Nor is there a solution to the equation ''x'' − φ(''x'') = 34, making 34 a
noncototient
In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function
In number theory ...
.
It is the third
Erdős–Woods number, following
22 and
16.
It is the ninth
Fibonacci number
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
and a companion
Pell number.
Since it is an odd-indexed Fibonacci number, 34 is a
Markov number.
34 is also the fourth
heptagonal number, and the first non-trivial
centered hendecagonal (11-gonal) number.
This number is also the
magic constant of
Queens Problem for
.
There are 34
topologically distinct convex heptahedra, excluding mirror images.
34 is the
magic constant of a
normal
magic square
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
,
and
magic octagram (see accompanying images); it is the only
for which magic constants of these
magic figures coincide.
See also
*
Rule 34
References
External links
Prime Curios! 34from the
Prime Pages
The PrimePages is a website about prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...
{{Integers, zero
Integers