34
This number is a composite.
34 = π(144). Note that 34 and 144 are Fibonacci numbers. No larger example of this type is known. Observed by G. L. Honaker, Jr. in 1999.
The first 34 odd numbers (concatenated) is prime.
[Das]
34 is the smallest number which can be expressed as the sum of two primes in four ways. [Murthy]
The smallest composite Fibonacci number whose sum of prime factors is a prime. [Gupta]
134 + 233 + 332 + ... + 323 + 332 + 341 is prime. [Patterson]
34 = 3 + 5 + 7 + 19 is the smallest number which is sum of distinct primes whose digits are odd. Note that all the odd digits are used, without repetition. [Capelle]
The smallest Fibonacci number f such that neither 6f - 1 nor 6f + 1 are prime. [Necula]
π(34)= !3 + !4, where !3 and !4 denotes subfactorial 3 and subfactorial 4 respectively. [Gupta]
34 = π(3!*4!). [Firoozbakht]
34!/34# ± 1 are twin primes. [Wesolowski]
π(34) = 3!! + 4!!. [Gupta]
There are 34 five-digit primes formed from the five odd digits. This means there's a Fibonacci number of Fibonacci-digit primes formed from the Fibonacci number of odd digits. [Silva and
Honaker]
The smallest balanced semiprime. [Silva]
The smallest integer that is the sum of two different perfect numbers. [Selassie]
7^34+34 is the smaller of only two non-titanic primes of form 7^n+n, (the other is for n=48). [Loungrides]
34 is a Fibonacci number F(9) that is simultaneously the sum of the squares of two consecutive primes (F(9) = 34 = 3^2 + 5^2) and the sum of the squares of two consecutive Fibonnaci's (F(9) = 34 = F(4)^2 + F(5)^2). [Rivera]
The number of the distinct-digit primes each consisting of
all of the odd digits. These are: 13597, 13759, 15739,
15937, 15973, 17359, 17539, 19753, 31957, 37159, 37591,
37951, 39157, 51973, 53197, 53719, 53791, 53917, 57139,
57193, 71359, 71593, 73951, 75193, 75391, 75913, 75931,
79153, 79531, 91573, 91753, 95317, 95713, 95731. [Loungrides]
The smallest semiprime that is the sum of two consecutive
prime squares (2*17=34=3^2+5^2). [Bajpai]
First occurrence of a run of exactly 34 consecutive integers with an odd number of prime factors has never been
found.
3^34+34 is the largest of three primes less than a googol of the form 3^n+n. [Loungrides]
Along with 34+1, the smallest pair of consecutive integers that are
both composite and reverse primes. [Gaydos]
Of all the primes produced by 34n + n - (n + 1)² for n = 1
- 34, only one is an isolated prime. Although the
twin primes produced are not distinct it is interesting to
note that there are two twin primes in the first three n values,
three from 5, i.e., 5/8, 8/13, 13/21 and twenty-one twin primes from thirty-four n
values, giving a ratio of primes produced exactly equal to
the Golden Ratio throughout the sequence. [Homewood]
The number of emirps less than 1000. [Loungrides]
The middle number in the first distinct semiprime treble cluster {33, 34, 35}. [Sela]