Infinitary Divisor

Author

Eric W. Weisstein
May 4, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/InfinitaryDivisor.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/InfinitaryDivisor.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

ExponentList[n_,factors_]:={#,IntegerExponent[n,#]}&/@factors
InfinitaryTable[n_]:=Module[{factors=First/@FactorInteger[n]},
{Power@@@#,Function[f,{f,IntegerDigits[f[[-1]],2,3]}]/@ExponentList[#,factors]
}&/@Divisors[n]
]

InfinitaryList[n_]:=Module[{factors=First/@FactorInteger[n]},
Last/@ExponentList[#,factors]&/@Divisors[n]
]

Examples

4

InfinitaryTable[8]//ColumnForm

InfinitaryDivisors[8]

9

InfinitaryTable[9]//ColumnForm

InfinitaryDivisors[9]

10

InfinitaryTable[10]//ColumnForm

InfinitaryDivisors[10]

12

InfinitaryTable[12]//ColumnForm

InfinitaryDivisors[12]

il=Transpose[InfinitaryList[12]]

BitOr[#,2]&/@il

BitOr[#,1]&/@il

Thread[BitOr[#,Last[#]]]&/@Transpose[InfinitaryList[12]]

Thread[Function[{f,g},BitOr[f,g]==g][#,Last[#]]]&/@Transpose[InfinitaryList[12]]

Transpose[Thread[Function[{f,g},BitOr[f,g]==g][#,Last[#]]]&/@Transpose[InfinitaryList[12]]]

Flatten[Position[%,_?(And@@#&),{1}]]

Divisors[12][[%]]

Enumeration

Table[{n,InfinitaryDivisors[n]},{n,15}]//ColumnForm

Length/@Table[InfinitaryDivisors[n],{n,30}]

Infinitary Divisor Function

Table[Plus@@InfinitaryDivisors[n],{n,10}]