I can't think of a specific number, but you also don't want to get a reputation. You should be strategic in trying to reach the hiring manager or HR manager. In addition to using LinkedIn to figure out whom to contact, you can also post on a networking group site or email list to see if any other member knows someone within the target company. Then contact that person directly and see if they would be kind enough to find out who the hiring manager is for a position and forward your email. A favor to you now begets a favor for them in the future when they are in job hunting mode.
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