# 34 (number)

 ← 33 34 35 →
Cardinalthirty-four
Ordinal34th
(thirty-fourth)
Factorization2 × 17
Divisors1, 2, 17, 34
Greek numeralΛΔ´
Roman numeralXXXIV
Binary1000102
Ternary10213
Senary546
Octal428
Duodecimal2A12

34 (thirty-four) is the natural number following 33 and preceding 35.

## In mathematics

34 is the ninth distinct semiprime,[1] with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form ${\displaystyle 2\times q}$. 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).[2]

The number 34 has an aliquot sum of 20, and is the seventh member in the aliquot sequence (34, 20, 22, 14, 10, 8, 7, 1, 0) that belongs to the prime 7-aliquot tree.

Its reduced totient and Euler totient values are both 16 (or 42 = 24).[3][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.[5] Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.[6]

It is the third Erdős–Woods number, following 22 and 16.[7]

It is the ninth Fibonacci number[8] and a companion Pell number.[9] Since it is an odd-indexed Fibonacci number, 34 is a Markov number.[10]

34 is also the fourth heptagonal number,[11] and the first non-trivial centered hendecagonal (11-gonal) number.[12]

This number is also the magic constant of ${\displaystyle n-}$Queens Problem for ${\displaystyle n=4}$.[13]

There are 34 topologically distinct convex heptahedra, excluding mirror images.[14]

34 is the magic constant of a ${\displaystyle 4\times 4}$ normal magic square,[15] and magic octagram (see accompanying images); it is the only ${\displaystyle n}$ for which magic constants of these ${\displaystyle n\times n}$ magic figures coincide.

34 is also:

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N. J. A. (ed.). "Sequence A056809". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than and equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
4. ^
5. ^ "Sloane's A005277 : Nontotients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. ^ "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
7. ^ "Sloane's A059756 : Erdős–Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-12-04.
8. ^ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
9. ^ "Sloane's A002203 : Companion Pell numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
10. ^ Weisstein, Eric W. "Markov Number". mathworld.wolfram.com. Retrieved 2020-08-21.
11. ^ "Sloane's A000566 : Heptagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
12. ^ Sloane, N. J. A. (ed.). "Sequence A069125 (Centered hendecagonal (11-gonal) numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-03.
13. ^ Sloane, N. J. A. (ed.). "Sequence A006003". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. ^ "Counting polyhedra". Numericana. Retrieved 2022-04-20.
15. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 53. ISBN 978-1-84800-000-1.
16. ^ Jason M. Highsmith, MD (2020-03-03). "Spinal Anatomy Center". SpineUniverse. Retrieved 2022-08-10.