Superabundant Number

Author

Eric W. Weisstein
December 23, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/SuperabundantNumber.nb.

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Table

v4.2

Timing[l=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10^6}]];
Flatten[Position[l,#,1,1]&/@Union[l]]
]

(l=Drop[FoldList[Plus,1,Length/@Split[FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10^6}]]]
],-1])//Timing

V7

(l=Most[Accumulate[Prepend[Length/@Split[FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10^6}]]],1]]])//Timing

(l=Most[Accumulate[Prepend[Length/@Split[FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10^7}]]],1]]])//Timing

MaxMemoryUsed[]/10^9.

(l=Most[Accumulate[Prepend[Length/@Split[FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10^8}]]],1]]])//Timing

MaxMemoryUsed[]/10^9.

Prime superabundants

Select[l,IntegerQ[Sqrt[#+1]]&]

Sqrt[#+1]&/@%

Comparison with highly composite numbers

s={1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,10080,15120,25200,27720,55440,110880,166320,277200,332640,554400,665280,720720,1441440,2162160,3603600,4324320,7207200,8648640,10810800};

h={1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,166320,221760,277200,332640,498960,554400,665280,720720,1081080,1441440};

Length/@{s,h}

s-Take[h,38]

Position[%,_?(#>0&)]

Transpose[{s,Take[h,38]}]//ColumnForm