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The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Ravenel's paper "Localization with respect to certain periodic homology theories" that $\langle S\rangle=\langle T(0)\rangle\vee\langle T(1)\rangle\vee\ldots\vee\langle T(n-1)\rangle\vee\langle F(n)\rangle$ where $F(n)$ is a finite spectrum of type n. Is it known whether or not $\langle S\rangle=\bigvee_n\langle T(n)\rangle$?
In the same vein, is it known whether or not $\bigvee_n\langle K(n)\rangle=\langle S\rangle$?
The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Ravenel's paper "Localization with respect to certain periodic homology theories" that $\langle S\rangle=\langle T(0)\rangle\vee\langle T(1)\rangle\vee\ldots\vee\langle T(n-1)\rangle\vee\langle F(n)\rangle$ where $F(n)$ is a finite spectrum of type n. Is it known whether or not $\langle S\rangle=\bigvee_n\langle T(n)\rangle$?