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Will Sendy handle a list of over 350k contacts?
Sendy can handle as much as your server can handle.
For example, if you have 350k contacts and you run Sendy on an under-powered shared server, Sendy will definitely not be able to handle 350k contacts very well.
If you have 350k contacts and you run Sendy on a VPS server with sufficient CPU and memory, Sendy will be able to handle 350k contacts very well.
What would you recommend as sufficient in a vm?
You can try an Amazon EC2 'Small' or 'Medium' instance (see EC2 instance types).
We have a VMware environment in the office so I'll throw a comparable instance together in there. If it works well I'll report back. Thanks!