Do Bernoulli Numbers Always Have Unique Prime Factors in Their Denominators?

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SUMMARY

The discussion centers on the conjecture regarding the uniqueness of prime factors in the denominators of Bernoulli numbers when expressed as irreducible fractions. It is established that the denominators of Bernoulli numbers, such as B_2 = 1/6, B_4 = -1/30, and B_{24} = -236364091/2730, do not contain powers of prime numbers. Specifically, it is noted that mathematician Srinivasa Ramanujan proved that the primes 2 and 3 appear only once in the factorization of these denominators, but the uniqueness of other primes remains unverified. The discussion references the Von Staudt–Clausen theorem for further context.

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Damidami
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Is this a known bernoulli number conjeture/theorem?:
The denominators of B_n (when expressed as an irreducible fraction), doesn't contain as a factor powers of prime numbers (ex. isn't divided by 5^2)

Example:

B_2 = 1/6
6 = 2*3

B_4 = -1/30
30 = 2*3*5

B_{24} = -236364091/2730
2730 = 2*3*5*7*13

I know Ramanujan proved that the denominator contain 2 and 3 as a factor one and only once, but I hadn't heard that any prime on the factorization of the denominator happens only once.
 
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Nevermind.

http://en.wikipedia.org/wiki/Von_Staudt%E2%80%93Clausen_theorem"

Aditional information is welcome.
 
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