Car will not start, have to turn key off and on until it starts.

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No clicking . no sounds at all. Have replaced starter with a new one, replaced ignition switch with a new one, had battery checked for correct voltage. All power accessories function when the key is turned to start position. Does not occur all of the time. Sometimes I have to recycle the start procedure 10X before it starts. 87000 miles on the vehicle Help, what do I replace next? Thank You for your HELP!