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34 (number)

FieldValue
number34
divisor1, 2, 17, 34

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

In mathematics

34 is the twelfth semiprime, with four divisors 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 numbers 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. Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.

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

It is the ninth Fibonacci number 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, and magic octagram (see accompanying images); it is the only n for which magic constants of these n \times n magic figures coincide.

References

References

  1. {{Cite OEIS. A001358. Semiprimes (or biprimes): products of two primes
  2. {{Cite OEIS. A056809. Numbers k such that k, k+1 and k+2 are products of two primes
  3. {{Cite OEIS. A000396. Perfect numbers k: k is equal to the sum of the proper divisors of k
  4. {{Cite OEIS. A02808. The composite numbers.
  5. {{Cite OEIS. A000010. Euler totient function phi(n): count numbers less than and equal to n and prime to n.
  6. {{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
  7. {{Cite OEIS. A005277. Nontotients
  8. {{Cite OEIS. A005278. Noncototients
  9. {{Cite OEIS. A059756. Erdős–Woods numbers
  10. {{Cite OEIS. A000045. Fibonacci numbers
  11. {{Cite OEIS. A002203. Companion Pell numbers
  12. Weisstein, Eric W.. "Markov Number".
  13. {{Cite OEIS. A000566. Heptagonal numbers
  14. {{Cite OEIS. A069125. Centered hendecagonal (11-gonal) numbers
  15. {{Cite OEIS. A006003
  16. "Counting polyhedra".
  17. Higgins, Peter. (2008). "Number Story: From Counting to Cryptography". Copernicus.
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