Near Multiperfects

                           Near-Multiperfects

A near-multiperfect has near-integer abundancy , i.e. , the sum of its
divisors leaves a small remainder when divided by some small multiple .
More specifically , a near-multiperfect is defined as any natural  n ,
with :

                sum-of-divisors ( n )   =   k * n  +  r
                           abs ( r )   <=   loge ( n )

loge is the log to the base e = natural log = the Napierian logarithm .
This definition includes the multiperfect .
A proper near-multiperfect is a near-multiperfect which is not
multiperfect .

Least Near-Multiperfects

This list of the least near-multiperfects excludes
a) primes ,
b) powers of 2 and
c) naturals of the form 6 times a prime .

          n    r
          6    0
         10   -2
         20    2
         28    0
         70    4
         88    4
        104    2
        110   -4
        120    0
        136   -2
        152   -4
        464    2
        496    0
        592   -6
        650    2
        672    0
        884   -4
       1155   -6
       1888    4
       1952    2
       2144   -4
       4030    4
       5830    4
       8128    0
       8384   -4
       8925    6
      11096    8
      17816    8
      18632   -4
      18904   -8
      30240    0
      32128    4
      32445    6
      32760    0
      32896   -2
      33664   -8
      45356    8
      70564   -8
      77744    8
      85936   -8
      91388    8
     100804   -8
     116624   -4
     128768    8
     130304    2
     133376  -10
     244036  -11
     254012    8
     388076    8
     391612   -8
     430272   12
     442365    6
     518656   10
     521728    4
     522752    2
     523776    0
     527872   -8
     528896  -10
    1090912   -8
    1848964    4
    2087936    8
    2102272   -6
    2178540    0
    2291936    8
    8378368    4
    8382464    2
    8394752   -4
    9928792   16
   11547352   16
   12026888  -16
   13174976    8
   13192768  -16
   15370304   -4
   16102808  -16
   17619844   -8
   17999992   16
   23569920    0
   26347688  -16
   29322008  -16
   29465852    8
   33501184   12
   33550336    0
   33653888  -16
   35021696    8
   45335936    8
   45532800    0
   73995392   -4
   89283592   16
  120888092    8
  134094848   14
  134193152    2
  142990848    0
  159030135   18
  169371008  -16
  173482552   16
  173631608  -16
  184773312   12
  260378492    8
  293947648  -16
  361702144   16
  381236216    8
  459818240    0
  536559616   18
  536920064   -4
  537051136  -12
  537116672  -16
  624032630   20
  775397948    8
  815634435   -6
  883927808  -16
 1081850752   16
 1113445430   20
 1379454720    0
 1476304896    0
 1550860550   20
 1631268870  -18
 1845991216   16
 2146926592   16
 2146992128   14
 2147516416   -2
 2147581952   -4
 2147713024   -8
 2147975168  -16
 2493705728  -16
 2586415095   18
 3381872252    8
 3915380170  -20
 4856970752    8
 5556840416  -16
 6077111050  -20
 6800228816    8
 6800695312   -8
 8589082624   12
 8589344768    8
 8589869056    0
 9796360330  -20
10828121356  -20
11097907192   16
12985220152   16
13092865928  -16
14182439040    0
21818579968   16
31998395520    0
34356723712   22
34357510144   16
34358296576   10
34359083008    4
34360131584   -4
34360655872   -8
34360918016  -10
42783299288  -16

count = 151
The multiperfects have remainder , r = 0 .
This list is complete through
43279999099 .

Walter Nissen
2008-01-21