login
A122943
Odd numbers n ordered by n/2^BigOmega(n), where BigOmega(n) is the number of prime divisors of n with repetition.
3
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
1,2
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).
FORMULA
A101695(n) = a(n) * 2^(n - BigOmega(a(n))). a(n) = A101695(n) / 2^A007814(A101695(n)) = A000265(A101695(n)).
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]]
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved