A122943 - OEIS

%I #13 Jul 14 2012 13:52:00

%S 1,3,9,5,27,7,15,81,21,11,45,25,13,243,63,33,135,17,35,75,19,39,729,

%T 23,189,49,99,405,51,105,55,225,57,29,117,31,125,65,2187,69,567,147,

%U 37,297,1215,153,77,315,41,165,675,85,171,43,87,175,351,91,93,375,47,95,195

%N Odd numbers n ordered by n/2^BigOmega(n), where BigOmega(n) is the number of prime divisors of n with repetition.

%C This is the limit of the sequence of largest odd factors of the k-almost primes as k -> infinity.

%C The location of 3^k in this sequence is A078843(k).

%H T. D. Noe, <a href="/A122943/b122943.txt">Table of n, a(n) for n = 1..1065</a>

%H <a href="/index/Al#almost primes, sequences related to">Almost primes, sequences related to</a>

%F A101695(n) = a(n) * 2^(n - BigOmega(a(n))). a(n) = A101695(n) / 2^A007814(A101695(n)) = A000265(A101695(n)).

%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}]]]]] (* from Eric Weisstein, Feb 07 2006 *); AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; f[n_] := Block[{ kap = AlmostPrime[20, n]}, kap / 2^IntegerExponent[ kap, 2]]; Array[f, 64] (* or *)

%t f[n_] := n/2^PrimeOmega[n]; Take[2 Ordering[ Table[ f[ 2n - 1], {n, 1100}]] - 1, 63] (* _Robert G. Wilson v_, Feb 08 2011 *)

%t f[n_] := n/2^PrimeOmega[n]; nn=9; t = Select[Table[{f[2 n - 1], 2 n - 1}, {n, 3^nn/2 + 1}], #[[1]] <= f[3^nn] &]; Transpose[Sort[t]][[2]]

%Y Cf. A001222, A101695, A078841, A078840, A000265, A007814.

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Oct 24 2006