Numbers with a prime factor > sqrt

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SUMMARY

This discussion focuses on categorizing non-prime numbers based on their prime factors relative to their square roots. Specifically, it distinguishes between numbers with a prime factor greater than their square root (category a) and those with all prime factors less than or equal to their square root (category b). The observed ratio of counts, Ca/Cb, appears to increase from 1.4 to 2.3, suggesting a trend worth investigating. The Dickman's rho function is highlighted as a relevant mathematical tool, providing an asymptotic estimate of smooth numbers, with \rho(2) indicating that approximately 30.69% of numbers are \sqrt{n}-smooth.

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  • Familiarity with asymptotic analysis and smooth numbers.
  • Knowledge of Dickman's rho function and its applications.
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  • Study the properties of Dickman's rho function in detail.
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  • Investigate the implications of prime factor distributions in non-prime numbers.
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This discussion is beneficial for mathematicians, number theorists, and students interested in advanced number theory concepts, particularly those studying prime factorization and smooth numbers.

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Suppose you divide all non-prime numbers in two categories, those which (a) have a prime factor greater than the square root of the number, and those which (b) don't, and all prime factors are less or equal than the square root.

Let Ca and Cb be the count of numbers in categories (a) and (b), resp. As you collect more numbers, a quick&dirty survey seems to indicate that the ratio Ca/Cb keeps growing (I don't know if converging), from 1.4 to 1.9 to 2.3... (Funny, actually I kind of imagined Cb to be bigger than Ca.)

What kind of math knowledge in number theory (or not) applies to the study of this? Any pointer, please?
 
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Dickman's rho function is an asymptotic estimate of the fraction of smooth numbers. In particular, \rho(2) is the asymptotic proportion of numbers that are \sqrt n-smooth. \rho(2)\approx30.69\% so your ratio should converge.

(The rho function, unlike your function, includes primes -- but they're asymptotically negligible.)
 

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