34 (number)
Template:Infobox number 34 (thirty-four) is the natural number following 33 and preceding 35.
In mathematicsEdit
34 is the twelfth semiprime,<ref>Template:Cite OEIS</ref> with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form <math>2 \times q</math>. 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).<ref>Template:Cite OEIS</ref>
34 is the sum of the first two perfect numbers 6 + 28,<ref>Template:Cite OEIS</ref> whose difference is its composite index (22).<ref>Template:Cite OEIS</ref>
Its reduced totient and Euler totient values are both 16 (or 42 = 24).<ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref> 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.<ref>Template:Cite OEIS</ref> Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.<ref>Template:Cite OEIS</ref>
It is the third Erdős–Woods number, following 22 and 16.<ref>Template:Cite OEIS</ref>
It is the ninth Fibonacci number<ref>Template:Cite OEIS</ref> and a companion Pell number.<ref>Template:Cite OEIS</ref> Since it is an odd-indexed Fibonacci number, 34 is a Markov number.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
34 is also the fourth heptagonal number,<ref>Template:Cite OEIS</ref> and the first non-trivial centered hendecagonal (11-gonal) number.<ref>Template:Cite OEIS</ref>
This number is also the magic constant of <math>n-</math>Queens Problem for <math>n=4</math>.<ref>Template:Cite OEIS</ref>
There are 34 topologically distinct convex heptahedra, excluding mirror images.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
34 is the magic constant of a <math>4 \times 4</math> normal magic square,<ref>Template:Cite book</ref> and magic octagram (see accompanying images); it is the only <math>n</math> for which magic constants of these <math>n \times n</math> magic figures coincide.
See alsoEdit
ReferencesEdit
External linksEdit
- Prime Curios! 34 from the Prime Pages