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 A276176 Consider the race between primes, semiprimes, 3-almost primes, ... k-almost 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A "k-almost prime" is a number which is the product of exactly k primes. Let pi_k(n) be the number of k-almost 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^(k-1))/(k-1)! for all k >= 0. Because of this fact, eventually the semiprimes will outnumber the primes; they do starting at 34. Likewise the 3-almost 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 4-almost and 3-almost 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 Giovanni Resta, Table of n, a(n) for n = 1..170 Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, page 211, (4); University of Michigan Historical Math Collection. FORMULA It seems plausible that 0.8 * log(A284411(m) - 1) <= log(a(n)) <= log(A284411(m)) in the instances where the overtaking concerns m-almost-primes and (m-1)-almost-primes. - 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 Cf. A243906, A273381, A274123, A358677, A359242 (restricted to squarefree numbers). Sequence in context: A374183 A022375 A072663 * A050905 A067571 A319486 Adjacent sequences: A276173 A276174 A276175 * A276177 A276178 A276179 KEYWORD hard,nonn AUTHOR Jonathan Vos Post and Robert G. Wilson v, Oct 09 2016 EXTENSIONS a(6)-a(22) from Giovanni Resta, Aug 17 2018 Name clarified by Peter Munn, Dec 31 2022 STATUS approved

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Last modified September 11 17:54 EDT 2024. Contains 375839 sequences. (Running on oeis4.)