A126281 - OEIS

%I #18 Apr 04 2015 09:12:02

%S 1,2,5,8,10,13,16,18,21,24,27,29,32,35,37,40,43,46,48,51,54,56,59,62,

%T 65,67,70,73

%N a(n) is the least m to satisfy the requirements of A052130.

%C A052130: For m very large, a(n) = number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity).

%C In observing the triangular array of A126279, the array T(k,n) defined as the k-th almost prime count of n-th power of two, it is noticed that the k-th term from the right converges to a fixed value beginning with the n-th power of two.

%C Will this sequence continue to match A117630: floor(n*log(3)/log(3/2)) ?

%H J. H. Smith, <a href="http://www.umass.edu/reports/temp77.html">Perfect Numbers</a>.

%t 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}]] ]]]; f[n_] := Block[{a = 0, m = n}, While[ b = AlmostPrimePi[m-n+1, 2^m]; b > a, m++; a = b]; m--; m]; Array[f, 24] (* _Eric W. Weisstein_, Feb 07 2006 *)

%Y Cf. A052130, A126279.

%K more,nonn

%O 1,2

%A _Robert G. Wilson v_, Dec 24 2006

%E a(25)-a(28) from _Robert G. Wilson v_, Sep 07 2012

%E Expression in comment corrected by _L. Edson Jeffery_, Apr 03 2015