A122943 Odd numbers n ordered by n/2^BigOmega(n), where BigOmega(n) is the number of prime divisors of n with repetition.

1, 3, 9, 5, 27, 7, 15, 81, 21, 11, 45, 25, 13, 243, 63, 33, 135, 17, 35, 75, 19, 39, 729, 23, 189, 49, 99, 405, 51, 105, 55, 225, 57, 29, 117, 31, 125, 65, 2187, 69, 567, 147, 37, 297, 1215, 153, 77, 315, 41, 165, 675, 85, 171, 43, 87, 175, 351, 91, 93, 375, 47, 95, 195

Offset: 1

Author

Franklin T. Adams-Watters, Oct 24 2006

Keywords

• nonn

Comments

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

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

Removing 1 and prime numbers from this sequence gives A374074. - Friedjof Tellkamp, Nov 27 2024

Links

• T. D. Noe, Table of n, a(n) for n = 1..1065
• Almost primes, sequences related to

Crossrefs

Cf. A001222, A101695, A078841, A078840, A000265, A007814, A374074.

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

Formula

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