This is not an answer but rather a comment to illustrate the observation.
Consider the canonical primefactor-decomposition of the denominators of that values. You'll easily observe the relevant patterns in them. For better reading I've done them in two columns, for zeta(1-1),zeta(1-2) ; zeta(1-3),zeta(1-4) ;...;zeta(1-k),zeta(1-(k+1)); for index k=1 to some m.
2 2^2 .3
1 2^3 .3 .5
1 2^2 .3^2 .7
1 2^4 .3 .5
1 2^2 .3 .11
1 2^3 .3^2 .5 .7 .13
1 2^2 .3 ----------------------------------------------------
1 2^5 .3 .5 .17
1 2^2 .3^3 .7 .19
1 2^3 .3 .5^2 .11
1 2^2 .3 .23
1 2^4 .3^2 .5 .7 .13
1 2^2 .3 ----------------------------------------------------
1 2^3 .3 .5 .29
1 2^2 .3^2 .7 .11 .31
1 2^6 .3 .5 .17
The rows which contain the horizontal lines denote the denominators which are exactly 12 (and do not only 12 as a factor). This patterns contain an obvious relation of the totient-value for the involved primefactors in relation to the index and are definite-ly described by the Clausen-von Staudt-theorem as mentioned in the other answer.
A tiny remark: Note, that the entry at index k=0 (zeta(1)) would contain all primefactors to infinite power.
