Why 2^^1*^k does not work?

Question

Table[2^^1*^k,{k,0,10}]

This gives an error, why is that?

Also this BaseForm[2 ** 1*^ #, 16] & /@ Range[10]

What is wrong here?

Answer 1

The number entry form base^^digits is only valid for explicit [0-9] digits in the place of both base and digits. You cannot write literal b^^1001 and then attempt to replace b as b^^1001 does not parse to this input form.

Likewise the number entry form m*^exp is only valid for explicit [0-9] digits in the place of both m and exp.

The combination of these entry forms does work, but possibly not in the manner you intend. (One must guess your intent.) For example 2^^1001*^12 inputs 36864, because 1001*^12 is first converted to 1001000000000000 and then 2^^1001000000000000 becomes 36864.

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Responding to the comment below, if the combined behavior is what you want you could "force" this evaluation by assembling a String and then converting it to an Expression using ToExpression:

Table[ToExpression[ "2^^1*^" <> ToString[k] ], {k, 0, 10}]

{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024}

This however makes one wonder why you would not simply write:

2^Range[0, 10]

{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024}

Or more generally:

Table[2^^1001 * 2^k, {k, 0, 10}]

{9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216}

Equivalent to:

 Table[ToExpression["2^^1001*^" <> ToString[k]], {k, 0, 10}]