34 (number)

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33 34 35
Cardinal thirty-four
Ordinal 34th
(thirty-fourth)
Factorization 2 × 17
Divisors 1, 2, 17, 34
Greek numeral ΛΔ´
Roman numeral XXXIV, xxxiv
Binary 1000102
Ternary 10213
Senary 546
Octal 428
Duodecimal 2A12
Hexadecimal 2216

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

Contents

In mathematics

34 is the twelfth semiprime, [1] with four divisors 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). [2]

Magic8star-sum34.svg
MagicSquare-AlbrechtDurer.png

34 is the sum of the first two perfect numbers 6 + 28, [3] whose difference is its composite index (22). [4]

Its reduced totient and Euler totient values are both 16 (or 42 = 24). [5] [6] 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. [7] Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient. [8]

It is the third Erdős–Woods number, following 22 and 16. [9]

It is the ninth Fibonacci number [10] and a companion Pell number. [11] Since it is an odd-indexed Fibonacci number, 34 is a Markov number. [12]

34 is also the fourth heptagonal number, [13] and the first non-trivial centered hendecagonal (11-gonal) number. [14]

This number is also the magic constant of Queens Problem for . [15]

There are 34 topologically distinct convex heptahedra, excluding mirror images. [16]

34 is the magic constant of a normal magic square, [17] and magic octagram (see accompanying images); it is the only for which magic constants of these magic figures coincide.

See also

References

  1. Sloane, N. J. A. (ed.). "SequenceA001358(Semiprimes (or biprimes): products of two primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  2. Sloane, N. J. A. (ed.). "SequenceA056809(Numbers k such that k, k+1 and k+2 are products of two primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  3. Sloane, N. J. A. (ed.). "SequenceA000396(Perfect numbers k: k is equal to the sum of the proper divisors of k)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  4. Sloane, N. J. A. (ed.). "SequenceA02808(The composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-06-02.
  5. Sloane, N. J. A. (ed.). "SequenceA000010(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.
  6. Sloane, N. J. A. (ed.). "SequenceA002322(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 On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  7. Sloane, N. J. A. (ed.). "SequenceA005277(Nontotients)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  8. Sloane, N. J. A. (ed.). "SequenceA005278(Noncototients)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  9. Sloane, N. J. A. (ed.). "SequenceA059756(Erdős–Woods numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  10. Sloane, N. J. A. (ed.). "SequenceA000045(Fibonacci numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  11. Sloane, N. J. A. (ed.). "SequenceA002203(Companion Pell numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  12. Weisstein, Eric W. "Markov Number". mathworld.wolfram.com. Retrieved 2020-08-21.
  13. Sloane, N. J. A. (ed.). "SequenceA000566(Heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  14. Sloane, N. J. A. (ed.). "SequenceA069125(Centered hendecagonal (11-gonal) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  15. Sloane, N. J. A. (ed.). "SequenceA006003(a(n) = n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  16. "Counting polyhedra". Numericana. Retrieved 2022-04-20.
  17. Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 53. ISBN   978-1-84800-000-1.