Example of a sequence with countable many cluster points Can someone give a concrete example of a sequence of reals that has countable infinite many cluster points ?

-- For $\,n=1,3,5,..., 2n-1,...\,$ , define $\,a_n=1\,$
-- For $\,n=2,6,10,...,2+4(n-1),...\,$ , define $\,a_n=2\,$
-- For $\,n=4,12,20,..,4+8(n-1),...\,$ , define $\,a_n=3\,$
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-- For $\,n=2^k,2^k+2^{k+1},...,2^k+2^{k+1}(n-1),...\,$ , define $\,a_n=k+1$
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Now just take the sequence $\,\left\{a_n\right\}_{n=1}^\infty\,$
