How to factorise a quantity obtained after summing?

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For your problem, since zeta(2n) has an explicit expression in terms of Bernoulli number I would rather go with

sage: [((-1)**(1+n//2)*2**(n-1) * bernoulli(n)/factorial(n)).factor() for n in range(2,20,2)]
[2^-1 * 3^-1,
 2^-1 * 3^-2 * 5^-1,
 3^-3 * 5^-1 * 7^-1,
 2^-1 * 3^-3 * 5^-2 * 7^-1,
 3^-5 * 5^-1 * 7^-1 * 11^-1,
 3^-6 * 5^-3 * 7^-2 * 11^-1 * 13^-1 * 691,
 2 * 3^-6 * 5^-2 * 7^-1 * 11^-1 * 13^-1,
 2^-1 * 3^-7 * 5^-4 * 7^-2 * 11^-1 * 13^-1 * 17^-1 * 3617,
 3^-9 * 5^-3 * 7^-3 * 11^-1 * 13^-1 * 17^-1 * 19^-1 * 43867]

Replace s.factor() by QQ(s).factor() to convert the symbolic expression to a rational and to factor that rational. Indeed, SR(1/6).factor() yields 1/6 while QQ(1/6).factor() yields 2^-1 * 3^-1.