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

1,1

Comments

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

Links

Table of n, a(n) for n=1..4.

Example

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

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 *)

Crossrefs

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

Sequence in context: A239775 A059072 A000459 * A217487 A173479 A173478

Adjacent sequences: A125285 A125286 A125287 * A125289 A125290 A125291

Keyword

nonn,hard,more

Author

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

Extensions

a(4) from Donovan Johnson, Nov 13 2010

Edited by Peter Munn, Jan 05 2023

Status

approved