

A276176


Consider the race between primes, semiprimes, 3almost primes, ... kalmost primes; sequence indicates when one overtakes another to give a new race leader.


4



2, 26, 31, 34, 15526, 151165506068, 151165506073, 151165506089, 151165506093, 151165506295, 151165506410, 151165506518, 151165506526, 151165506658, 151165506665, 151165506711, 151165506819, 151165506970, 151165506994, 151165507256, 151165507259, 151165507265
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OFFSET

1,1


COMMENTS

A "kalmost prime" is a number which is the product of exactly k primes.
Let pi_k(n) be the number of kalmost primes less than or equal to n. In 1909, on page 211 of the Handbuch, Edmund G. H. Landau stated that pi_k(n) ~ (n/log n)*(log log n^(k1))/(k1)! for all k >= 0.
Because of this fact, eventually the semiprimes will outnumber the primes; they do starting at 34. Likewise the 3almost primes will outnumber the semiprimes and they do starting at 15526.
The terms from a(6) = 151165506068 to a(170) = 151165607026 correspond to counts of 4almost and 3almost primes overtaking each other multiple times.  Giovanni Resta, Aug 17 2018


REFERENCES

Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, or Chelsea Publishing, NY 1953, or Vol. 1, Teubner, Leipzig; third edition: Chelsea, New York 1974.


LINKS



FORMULA

It seems plausible that 0.8 * log(A284411(m)  1) <= log(a(n)) <= log(A284411(m)) in the instances where the overtaking concerns malmostprimes and (m1)almostprimes.  Peter Munn, Aug 03 2023


EXAMPLE

a(1) = 2 since beginning with the natural numbers (A000027) the race is even with no group in the lead. But at 2, we encounter our first member (1 is unity and is not a member of any group here) which is a prime and therefore the primes take the lead with 2.
a(2) = 34 which is a semiprime. pi_1(34) = 11 and pi_2(34) = 12. This is the first time that the semiprimes overtake the primes.


MATHEMATICA

k = 1; lst = {}; tf = 0; p1 = 0; p2 = 0; While[k < 100001, If[PrimeOmega@k == 1, p1++]; If[PrimeOmega@k == 2, p2++]; If[p1 > p2 && tf == 0, tf++; AppendTo[lst, k]]; If[p2 > p1 && tf == 1, tf; AppendTo[lst, k]]; k++]; lst
(* cross check using *) AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[ Array[a, k  1]]]  a[k  1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i  1], PrimePi[(n/Times @@ Prime[Array[a, i  1]])^(1/(k  i + 1))]}, {i, k  1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)
(* as an example *) AlmostPrimePi[2, 15526] => 3986 whereas AlmostPrimePi[3, 15526] => 3987.


CROSSREFS



KEYWORD

hard,nonn


AUTHOR



EXTENSIONS



STATUS

approved



