1,1

Pi_n(a(n)/a(n): 0.66667, 0.40000, 0.25801, 0.2145966850

3=3, 10=2*5, 9837=3*3*1093 & 251179584927=3*271*1753*176243.

3 is the second prime, 10 is the fourth semiprime, 9837 is the trieneprime and 251179584927 is the 4-almost prime.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(1)=3 because Pi(2)/2=1/2 < Pi(3)/3=2/3 > Pi(5)/5=3/5.

a(2)=10 because Pi_2(9)/9=1/3 < Pi_2(10)/10=2/5 > Pi_2(14)/14=5/14; Pi_2(10)/10 = Pi_2(15)/15 but 10 < 15.

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 (eww(AT)wolfram.com) Feb 07 2006 *)

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 4; c = r = 0; Do[If[fQ@n, c++ ]; If[c/n > r, Print[n]; r = c/n], {n, 10^6}]

Cf. A006880, A066265, A109251, A114106, A114453.

Sequence in context: A103156 A012865 A006273 this_sequence A051498 A092528 A069604

Adjacent sequences: A117523 A117524 A117525 this_sequence A117527 A117528 A117529

bref,nonn

Martin Raab (raab-martin(AT)gmx.de) and Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 25 2006

a(4) is in the neighborhood of 2e11.