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Is there a function for the number of integers below n having only prime factors greater than or equal to p? For example, how do i determine the number of integers below 1000 only having prime factors greater than or equal to 7?

The only approach I can think of is:

1000/7 - 1000/(2*7)- 1000/(2*3*7) - 1000/(2*3*5*7)

Which can be re-expressed as:

1000/7 * (1 - 1/2 - 1/2*3 - 1/2*3*5) = 42 (which is incorrect).

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closed as off-topic by Myshkin, Alexey Ustinov, Jeremy Rickard, Wolfgang, Marco Golla Nov 18 '15 at 15:05

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Myshkin, Alexey Ustinov, Jeremy Rickard, Wolfgang, Marco Golla
If this question can be reworded to fit the rules in the help center, please edit the question.

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In general, in Multiplicative Number Theory you can't hope to obtain (relatively easy) "functions" for the kind of things you asked, but only asymptotic estimates. For your problem see Chapter III.6 of G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory.

P.S. I do not think this question is suitable for MO.

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  • $\begingroup$ Thanks - that's helpful. Apologies - didn't realise this site was research-level stuff. $\endgroup$ – user82925 Nov 18 '15 at 20:16

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