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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 2 \times q. 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 42 = 24). 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 n- Queens Problem for n=4. There are 34 topologically distinct convex heptahedra, excluding mirror images. 34 is the magic constant of a 4 \times 4 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 n for which magic constants of these n \times n magic figures coincide.


See also

* Rule 34


References


External links


Prime Curios! 34
from 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