*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.

    at indexes      at indexes                  
    k=1,3,5,7...    k=2,4,6,8,...             
    -------------   ---------------------------                           
      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 contain $12$ 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.