34 (number)

34 is the twelfth semiprime,{{Cite OEIS|A001358|Semiprimes (or biprimes): products of two primes}} with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form $2\; \backslash times\; q$. 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).{{Cite OEIS|A056809|Numbers k such that k, k+1 and k+2 are products of two primes}}

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34 is the sum of the first two perfect numbers 6 + 28,{{Cite OEIS|A000396|Perfect numbers k: k is equal to the sum of the proper divisors of k}} whose difference is its composite index (22).{{Cite OEIS |A02808 |The composite numbers. |access-date=2024-06-02 }}

Its reduced totient and Euler totient values are both 16 (or 4² = 2⁴).{{Cite OEIS |A000010 |Euler totient function phi(n): count numbers less than and equal to n and prime to n. |access-date=2023-09-11 }}{{Cite OEIS|A002322|Reduced totient function psi(n): least k such that x^k congruent to 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n}} 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.{{Cite OEIS|A005277|Nontotients}} Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.{{Cite OEIS|A005278|Noncototients}}

It is the third Erdős–Woods number, following 22 and 16.{{Cite OEIS|A059756|Erdős–Woods numbers}}

It is the ninth Fibonacci number{{Cite OEIS|A000045|Fibonacci numbers}} and a companion Pell number.{{Cite OEIS|A002203|Companion Pell numbers}}

Since it is an odd-indexed Fibonacci number, 34 is a Markov number.{{Cite web|last=Weisstein|first=Eric W.|title=Markov Number|url=https://mathworld.wolfram.com/MarkovNumber.html|access-date=2020-08-21|website=mathworld.wolfram.com|language=en}}

34 is also the fourth heptagonal number,{{Cite OEIS|A000566|Heptagonal numbers}} and the first non-trivial centered hendecagonal (11-gonal) number.{{Cite OEIS|A069125|Centered hendecagonal (11-gonal) numbers}}

This number is also the magic constant of $n-$Queens Problem for $n=4$.{{Cite OEIS|A006003|2=a(n) = n*(n^2 + 1)/2}}

There are 34 topologically distinct convex heptahedra, excluding mirror images.{{Cite web|url=http://www.numericana.com/data/polycount.htm|website=Numericana|title=Counting polyhedra|access-date=2022-04-20}}

34 is the magic constant of a $4\; \backslash times\; 4$ normal magic square,{{cite book |title=Number Story: From Counting to Cryptography |last=Higgins |first=Peter |year=2008 |publisher=Copernicus |location=New York |isbn=978-1-84800-000-1 |page=53 }} and magic octagram (see accompanying images); it is the only $n$ for which magic constants of these $n\; \backslash times\; n$ magic figures coincide.

• Rule 34

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• [http://primes.utm.edu/curios/page.php/34.html Prime Curios! 34] from the Prime Pages

{{Integers|zero}}

Category:Integers