A125288 a(n) = least integer k such that for all integers m greater than k, 2*Pi(n,m) is greater than Pi(n,2*m).

10, 297, 49650, 180701087317

Offset: 1

Author

Jonathan Vos Post and Robert G. Wilson v, Jan 22 2007

Keywords

• nonn
• hard
• more

Comments

Pi(n, m) is the number of integers <= m that have n prime factors counting multiplicity, also known as n-almost-primes (A078840).

Examples

			a(1) = 10 since the first term relates to 1-almost-primes, which are the primes themselves; and there are 4 primes <= 10, and 2*4 = 8 primes <= 2*10 = 20; but for m = 11 and all larger integers, the number of primes <= 2*m is less than twice the number of primes <= m. - _Peter Munn_, Dec 23 2022
		

Crossrefs

Cf. A078840, A092097, A117526, A126279, A125149.

Programs

Mathematica

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 *)

Extensions

a(4) from Donovan Johnson, Nov 13 2010

Edited by Peter Munn, Jan 05 2023