If I have a large number, for example,
2^11 is there code to list a table of positive integers less than this number which are not of the form
p^k *j where $p$ is prime and $j\not=p$ is either prime or 1?
Update: The restriction
p^k * j < 2^11 forces (in case j=1)
p^k < 2^11 for each given p. So for each $p$, the largest $k$ (let's denote it $k_p$) for which $p^{k_p}<2^{11}$ is $k_p=$
Floor[11 Log[2] / Log[p]] 
max = 2^11; ps = Prime[Range[PrimePi[max - 1]]]; ks = Range /@ Floor[Log[ps, N[max - 1]]]; js = # /. (# -> Append[ps[[;; First[FirstPosition[ps, n_ /; n > #, -1, {1}]]]], 1] & /@ DeleteDuplicates[#]) &[Floor[(max - 1) ps^-1]]; Complement[Range[max - 1], Flatten[MapThread[Outer[Times, #, #2] &, {ps^ks, js}]]] {1, 30, 36, 42, 60, 66, 70, 72, 78, 84, 90, 100, 102, 105, 108, 110 .......
1 is included because i didn't use $k=0$. If thats wrong it should be ks = Range[0, #]& /@ Floor[Log[ps, N[max - 1]]];
If I understand your question correctly, then the form you describe can be tested with the following function:
PKJFormQ[k_Integer] := With[ {factors = FactorInteger[k]}, And[ Length[factors] <= 2, Length@Cases[factors[[All, 2]], Except[1], {1}] <= 1]]; SetAttributes[PKJFormQ, Listable]; (I.e., there are at most 2 prime factors and either one or both of the prime factors has an exponent of 1.) You can then use any number of given Mathematica tools (like Pick or Reap) to do the rest; here are a couple examples on the limited range of up to 2^6:
First@Last@Reap@Do[If[!PKJFormQ[k], Sow[k]], {k, 2^6}] {30, 36, 42, 60}
Pick[#, PKJFormQ[#], False]&@Range[2^6] {30, 36, 42, 60}
The full list is a bit long for 2^11, so I won't paste it here, but it takes no time to run:
Length@Pick[#, PKJFormQ[#], False]&@Range[2^11] 773
Complement eliminated identical coefficients; now fixed. $\endgroup$
k? $\endgroup$Pick[#, Not[PrimeQ[FactorInteger[#, 2][[-1, 1]]]] & /@ #] &@ Range[2^11]? $\endgroup$