Let an m-prime be a square-free number with m prime divisors. Also let the number of t-primes < N be represented as #(t-prime){N} (t and N being positive elements of integers). Is the following true? Given a fixed integer N ; #(2-prime){N} >= #(3-prime){N} >= #(4-primes){N} >=....etc.??


The statement is false -- although it holds for quite some time.

If $c_t(n)$ denotes the number of $t$-primes smaller than or equal to $n$, we have $$c_2(1279789)<c_3(1279789)$$ and also $$c_1(58)<c_2(58)$$ (the latter wasn't part of the conjectured series of inequalities, though).

Based on a quick non-rigorous reasoning (which could be wrong, of course), it seems that neither of the the inequalities can hold indefinitely: apparently, the count of $(t+1)$-primes starts as smaller than the count of $t$-primes, but it eventually catches up and overtakes the other (and this "eventually" happens at different point for each inequality). In order to see why, consider the estimate given in Wikipedia: $$\pi_t(n)\sim \frac{n}{\log n} \frac{\left(\log \log n\right)^{t-1}}{(t-1)!}$$ Although the result in Wikipedia is stated for numbers with $t$ prime factors when counting multiplicity, it should also be true for numbers with $t$ distinct prime factors and also for square-free numbers with $t$ distinct prime factors; which is precisely our $c_t(n)$. Since the expression on right-hand side grows asymptotically faster when $t$ is greater (higher exponent at $\log \log n$), it eventually outgrows the one with smaller $t$. However, the speed of growth is proportional to double logarithm of $n$, so it might be a very slow process...

  • $\begingroup$ If Ct(n) is the number of t-primes<=n then is {C2(n)+C4(n)+C6(n)+...}< {C1(n)+C3(n)+....} ? $\endgroup$ – user128932 Apr 18 '14 at 2:18
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    $\begingroup$ Nope, the inequality between sums breaks up at $n=96$: $C_1(96)=24$, $C_2(96)=30$ and $C_3(96)=5$. The expected asymptotic behaviour is even more difficult to predict in this case, though. $\endgroup$ – Peter Košinár Apr 18 '14 at 6:01
  • $\begingroup$ If Ct(n) is the number of t primes < = n then is {C3(n) + C5(n) +..} > {C2(n) + C4(n) +..} ? $\endgroup$ – 201044 Sep 14 '15 at 4:14
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    $\begingroup$ @201044 Clearly no: The sum on the right-hand side becomes non-zero first ($C_2(6)=1$, whereas the left-hand side needs to wait until $C_3(30)=1$ to even get off the zero. $\endgroup$ – Peter Košinár Oct 5 '15 at 22:13
  • $\begingroup$ So if {C3(N) + C5(N)+...} < { C2(N) + C4(N) +...} for N <= 30 is this true for any N > 30? $\endgroup$ – 201044 Mar 18 '16 at 23:52

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